Sample records for nonlinear iterative method

This paper describes geometrical essentials of some iterationmethods (e.g. Newton iteration,secant line method,etc.) for solving nonlinear equations and advances some geomet-rical methods of iteration that are flexible and efficient.

This book focuses on the approximation of nonlinear equations using iterativemethods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterativemethods and shows that selection of the most efficient and robust iterativemethod for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...

This paper discusses the accelerating iterativemethods for solving the implicit scheme of nonlinear parabolic equations. Two new nonlineariterativemethods named by the implicit-explicit quasi-Newton (IEQN) method and the derivative free implicit-explicit quasi-Newton (DFIEQN) method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iterationmethod can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov (JFNK) method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear (inner) iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iterationmeth-ods are presented in confirmation of the theory and comparison of the performance of these methods.

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iterationmethods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.

Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterativemethods for solving nonlinear equations. The ninth order convergent Newton-type iterativemethod is made derivative free to obtain seventh-order convergent Newton-type iterativemethod. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterativemethods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.

We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlineariterativemethods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlineariterativemethods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.

The performance of elastic-plastic mixed-iterative analysis is examined through a set of convergence studies. Membrane and bending behaviors are tested using 4-node quadrilateral finite elements. The membrane result is excellent, which indicates the implementation of elastic-plastic mixed-iterative analysis is appropriate. On the other hand, further research to improve bending performance of the method seems to be warranted.

In this paper, we present and analyze a family of fifth-order iterativemethods free from second derivative for solving nonlinear equations. It is established that the family of iterativemethods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

In this paper, we shall use the variational iterationmethod to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV-MKdV equation and Camassa-Holm equation. The variational iterationmethod is superior than the other non-linearmethods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.

Full Text Available A modification of the variational iterationmethod (VIM for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE obtained using the traditional variational iterationmethod give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iterationmethod. Illustrative examples are presented to show the validity of this modification.

Full Text Available A family of eighth-order iterativemethods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency index of the developed methods is 1.682. Some numerical examples illustrate that the algorithms are more efficient and performs better than the other methods.

Nonlinear state-delayed optimal control problems have complex nonlinear characters. To solve this complex nonlinear problem, an iterative symplectic pseudospectral method based on quasilinearization techniques, the dual variational principle and pseudospectral methods is proposed in this paper. First, the proposed method transforms the original nonlinear optimal control problem into a series of linear quadratic optimal control problems. Then, a symplectic pseudospectral method is developed to solve these converted linear quadratic state-delayed optimal control problems. Coefficient matrices in the proposed method are sparse and symmetric since the dual variational principle is used, which makes the proposed method highly efficient. Converged numerical solutions with high precision can be obtained after a few iterations due to the benefit of the local pseudospectral method and quasilinearization techniques. In the numerical simulations, other numerical methods were used for comparisons. The numerical simulation results show that the proposed method is highly accurate, efficient and robust.

Full Text Available A homotopy method is presented for the construction of frozen Jacobian iterativemethods. The frozen Jacobian iterativemethods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterativemethod. We embedded parameters in the iterativemethods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterativemethods, for solving weakly nonlinear systems of equations. Further iterativemethods are also proposed for solving general systems of nonlinear equations.

Full Text Available The variational iterationmethod (VIM is applied to solve the boundary layer problem of magnetohydrodynamic flow over a nonlinear stretching sheet. The combination of the VIM and the Padé approximants is shown to be a powerful method for solving two-point boundary value problems consisting of systems of nonlinear differential equations. And the comparison of the obtained results with other available results shows that the method is very effective and convenient for solving boundary layer problems.

In this paper we presented a convergence condition of parallel dynamic iterationmethods for a nonlinear system of differential-algebraic equations with a periodic constraint.The convergence criterion is decided by the spectral expression of a linear operator derivedfrom system partitions. Numerical experiments given here confirm the theoretical work ofthe paper.

Full Text Available In this paper, we develop new families of sixth-order methods for solving simple zeros of non-linear equations. These methods are constructed such that the convergence is of order six. Each member of the families requires two evaluations of the given function and two of its derivative per iteration. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples.

In this paper, we present an iterativemethod for the regularization of ill-posed, nonlinear problems. The approach is based on the Bregman projection onto stripes the width of which is controlled by both the noise level and the structure of the operator. In our investigations, we follow (Lorenz et al 2014 SIAM J. Imaging Sci. 7 1237-62) and extend the respective method to the setting of nonlinear operators. Furthermore, we present a proof for the regularizing properties of the method.

Full Text Available When we need to determine the solution of a nonlinear equation there are two options: closed-methods which use intervals that contain the root and during the iterative process reduce the size of natural way, and, open-methods that represent an attractive option as they do not require an initial interval enclosure. In general, we know open-methods are more efficient computationally though they do not always converge. In this paper we are presenting a divergence case analysis when we use the method of fixed point iteration to find the normal height in a rectangular channel using the Manning equation. To solve this problem, we propose applying two strategies (developed by authors that allow to modifying the iteration function making additional formulations of the traditional method and its convergence theorem. Although Manning equation is solved with other methods like Newton when we use the iterationmethod of fixed-point an interesting divergence situation is presented which can be solved with a convergence higher than quadratic over the initial iterations. The proposed strategies have been tested in two cases; a study of divergence of square root of real numbers was made previously by authors for testing. Results in both cases have been successful. We present comparisons because are important for seeing the advantage of proposed strategies versus the most representative open-methods.

A nonlineariterative scheme developed to reduce the computing time of the AFEN method was tested and applied to two benchmark problems. The new nonlinearmethod for the AFEN method is based on solving two-node problems and use of two nonlinear correction factors at every interface instead of one factor in the conventional scheme. The use of two correction factors provides higher-order accurate interface fluxes as well as currents which are used as the boundary conditions of the two-node problem. The numerical results show that this new method gives exactly the same solution as that of the original AFEN method and the computing time is significantly reduced in comparison with the original AFEN method. 7 refs., 1 fig., 1 tab. (Author)

The authors report on some iterativemethods which they have tested for use in combustion simulations. In particular, they have developed a code to solve zero Mach number reacting flow equations with complex reaction and diffusion physics. These equations have the form of a nonlinear parabolic system coupled with constraints. In semi-discrete form, one obtains DAE`s of index two or three depending on the number of spatial dimensions. The authors have implemented a fourth order (fully implicit) BDF method in time, coupled with a suite of fourth order explicit and implicit spatial difference approximations. Most codes they know of for simulating reacting flows use a splitting strategy to march in time. This results in a sequence of nonlinear systems to solve, each of which has a simpler structure than the one they are faced with. The rapid and robust solution of the coupled system is the essential requirement for the success of their approach. They have implemented and analyzed nonlinear generalizations of conjugate gradient-like methods for nonsymmetric systems, including CGS and the quasi-Newton based method of Eirola and Nevanlinna. They develop a general framework for the nonlinearization of linear methods in terms of the acceleration of fixed-point iterations, where the latter is assumed to include the {open_quote}preconditioning{open_quote}. Their preconditioning is a single step of a split method, using lower order spatial difference approximations as well as simplified (Fickian) approximations of the diffusion physics.

Full Text Available This paper develops a modified variational iterationmethod coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs. The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

An iterative solution method for fully nonlinear boundary value problems governing self-similar flows with a free boundary is presented. Specifically, the method is developed for application to water entry problems, which can be studied under the assumptions of an ideal and incompressible fluid with negligible gravity and surface tension effects. The approach is based on a pseudo time stepping procedure, which uses a boundary integral equation method for the solution of the Laplace problem governing the velocity potential at each iteration. In order to demonstrate the flexibility and the capabilities of the approach, several applications are presented: the classical wedge entry problem, which is also used for a validation of the approach, the block sliding along an inclined sea bed, the vertical water entry of a flat plate and the ditching of an inclined plate. The solution procedure is also applied to cases in which the body surface is either porous or perforated. Comparisons with numerical or experimental d...

In this article, He’s variational iterationmethod (VIM), is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely...... and nonlinear problems. It is predicted that VIM can be widely applied in engineering....

Diffuse Optical Tomography (DOT) has become an emerging non-invasive technology, and has been widely used in clinical diagnosis. Functional near-infrared (FNIR) is one of the important applications of DOT. However, FNIR is used to reconstruct two-dimensional (2D) images for the sake of good spatial and temporal resolution. In this paper we propose a multiple-input and multiple-output (MIMO) based data extraction algorithm method in order to increase the spatial and temporal resolution. The non-lineariterativemethod is used to reconstruct better resolution images layer by layer. In terms of theory, the simulation results and original images are nearly identical. The proposed reconstruction method performs good spatial resolution, and has a depth resolutions capacity of three layers.

Full Text Available This paper studies nonlinear vibration mechanism of hard coating thin plate based on macroscopic vibration theory and proposes finite element iterationmethod (FEIM to theoretically calculate its nature frequency and vibration response. First of all, strain dependent mechanical property of hard coating is briefly introduced and polynomial method is adopted to characterize the storage and loss modulus of coating material. Then, the principle formulas of inherent and dynamic response characteristics of the hard coating composite plate are derived. And consequently specific analysis procedure is proposed by combining ANSYS APDL and self-designed MATLAB program. Finally, a composite plate coated with MgO + Al2O3 is taken as a study object and both nonlinear vibration test and analysis are conducted on the plate specimen with considering strain dependent mechanical parameters of hard coating. Through comparing the resulting frequency and response results, the practicability and reliability of FEIM have been verified and the corresponding analysis results can provide an important reference for further study on nonlinear vibration mechanism of hard coating composite structure.

In this article, an application of He's variational iterationmethod is proposed to approximate the solution of a nonlinear fractional differential equation with Riemann-Liouville's fractional derivatives. Also, the results are compared with those obtained by Adomian's decomposition method and truncated series method. The results reveal that the method is very effective and simple.

We present a fixed-point iterativemethod for solving systems of nonlinear equations. The convergence theorem of the proposed method is proved under suitable conditions. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.

Full Text Available In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i Modified Adomian decomposition method (MADM, (ii Variational iterationmethod (VIM, (iii Homotopy analysis method (HAM and (iv Modified homotopy perturbation method (MHPM. The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.

Full Text Available In this paper, we have achieved high order solution of a three dimensional nonlinear diffusive-convective problem using modified variational iterationmethod. The efficiency of this approach has been shown by solving two examples. All computational work has been performed in MATHEMATICA.

The author gives a hybrid method for the iterative solution of linear systems of equations Ax = b, where the matrix (A) is nonsingular, sparse and nonsymmetric. As in a method developed by Starke and Varga the method begins with a number of steps of the Arnoldi method to produce some information on the location of the spectrum of A. This method then switches to an iterativemethod based on the Faber polynomials for an annular sector placed around these eigenvalue estimates. The Faber polynomials for an annular sector are used because, firstly an annular sector can easily be placed around any eigenvalue estimates bounded away from zero, and secondly the Faber polynomials are known analytically for an annular sector. Finally the author gives three numerical examples, two of which allow comparison with Starke and Varga`s results. The third is an example of a matrix for which many iterativemethods would fall, but this method converges.

In this research, the variational iterationmethod as an approximate analytical method is utilized to overcome some inherent limitations arising as uncontrollability to the nonzero endpoint boundary conditions and is used to solve some examples in the field of heat transfer. The available exact s...

A general iterative-restoration principle is introduced to facilitate the implementation of nonlinear optical processors. The von Neumann convergence theorem is generalized to include nonorthogonal subspaces which can be reduced to a special orthogonal projection operator by applying an orthogonality condition. This principle is shown to permit derivation of the Jacobi algorithm, the recursive principle, the van Cittert (1931) deconvolution method, the iteration schemes of Gerchberg (1974) and Papoulis (1975), and iteration schemes using two Fourier conjugate domains (e.g., Fienup, 1981). Applications to restoring the image of a double star and division by hard and soft zeros are discussed, and sample results are presented graphically.

A class of El Nino atmospheric physics oscillation model is considered. The El Nino atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and westem Pacific anomaly patterns. An El Nino atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iterationmethod, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the seaair oscillation for El Nino atmospheric physics model can be analyzed. El Nino is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.

This is a project in the field of fundamental research on numerical methods for solving the particle transport equation. Numerous practical problems require to use unstructured meshes, for example, detailed nuclear reactor assembly-level calculations, large-scale reactor core calculations, radiative hydrodynamics problems, where the mesh is determined by hydrodynamic processes, and well-logging problems in which the media structure has very complicated geometry. Currently this is an area of very active research in numerical transport theory. main issues in developing numerical methods for solving the transport equation are the accuracy of the numerical solution and effectiveness of iteration procedure. The problem in case of unstructured grids is that it is very difficult to derive an iteration algorithm that will be unconditionally stable.

Full Text Available A new algorithm called multistep reproducing kernel Hilbert space method is represented to solve nonlinear oscillator’s models. The proposed scheme is a modification of the reproducing kernel Hilbert space method, which will increase the intervals of convergence for the series solution. The numerical results demonstrate the validity and the applicability of the new technique. A very good agreement was found between the results obtained using the presented algorithm and the Runge-Kutta method, which shows that the multistep reproducing kernel Hilbert space method is very efficient and convenient for solving nonlinear oscillator’s models.

Much research has been devoted to complex-variable optimization problems due to their engineering applications. However, the complex-valued optimization method for solving complex-variable optimization problems is still an active research area. This paper proposes two efficient complex-valued optimization methods for solving constrained nonlinear optimization problems of real functions in complex variables, respectively. One solves the complex-valued nonlinear programming problem with linear equality constraints. Another solves the complex-valued nonlinear programming problem with both linear equality constraints and an ℓ₁-norm constraint. Theoretically, we prove the global convergence of the proposed two complex-valued optimization algorithms under mild conditions. The proposed two algorithms can solve the complex-valued optimization problem completely in the complex domain and significantly extend existing complex-valued optimization algorithms. Numerical results further show that the proposed two algorithms have a faster speed than several conventional real-valued optimization algorithms.

The class of overlapping Schwarz algorithms has been extensively studied for linear elliptic finite element problems. In this presentation, the author considers the solution of systems of nonlinear algebraic equations arising from the finite element discretization of some nonlinear elliptic equations. Several overlapping Schwarz algorithms, including the additive and multiplicative versions, with inexact Newton acceleration will be discussed. The author shows that the convergence rate of the Newton`s method is independent of the mesh size used in the finite element discretization, and also independent of the number of subdomains into which the original domain in decomposed. Numerical examples will be presented.

This graduate-level text examines the practical use of iterativemethods in solving large, sparse systems of linear algebraic equations and in resolving multidimensional boundary-value problems. Assuming minimal mathematical background, it profiles the relative merits of several general iterative procedures. Topics include polynomial acceleration of basic iterativemethods, Chebyshev and conjugate gradient acceleration procedures applicable to partitioning the linear system into a "red/black" block form, adaptive computational algorithms for the successive overrelaxation (SOR) method, and comp

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Full Text Available This paper seeks to use the proposed residual correction method in coordination with the monotone iterative technique to obtain upper and lower approximate solutions of singularly perturbed non-linear boundary value problems. First, the monotonicity of a non-linear differential equation is reinforced using the monotone iterative technique, then the cubic-spline method is applied to discretize and convert the differential equation into the mathematical programming problems of an inequation, and finally based on the residual correction concept, complex constraint solution problems are transformed into simpler questions of equational iteration. As verified by the four examples given in this paper, the method proposed hereof can be utilized to fast obtain the upper and lower solutions of questions of this kind, and to easily identify the error range between mean approximate solutions and exact solutions.

A sparse nonlinear electromagnetic imaging scheme is proposed for reconstructing dielectric contrast of investigation domains from measured fields. The proposed approach constructs the optimization problem by introducing the sparsity constraint to the data misfit between the scattered fields expressed as a nonlinear function of the contrast and the measured fields and solves it using the nonlineariterative shrinkage thresholding algorithm. The thresholding is applied to the result of every nonlinear Landweber iteration to enforce the sparsity constraint. Numerical results demonstrate the accuracy and efficiency of the proposed method in reconstructing sparse dielectric profiles.

At the ship's design, the first step of the hull structural assessment is based on the longitudinal strength analysis, with head wave equivalent loads by the ships' classification societies’ rules. This paper presents an enhancement of the longitudinal strength analysis, considering the general case of the oblique quasi-static equivalent waves, based on the own non-lineariterative procedure and in-house program. The numerical approach is developed for the mono-hull ships, without restrictions on 3D-hull offset lines non-linearities, and involves three interlinked iterative cycles on floating, pitch and roll trim equilibrium conditions. Besides the ship-wave equilibrium parameters, the ship's girder wave induced loads are obtained. As numerical study case we have considered a large LPG liquefied petroleum gas carrier. The numerical results of the large LPG are compared with the statistical design values from several ships' classification societies’ rules. This study makes possible to obtain the oblique wave conditions that are inducing the maximum loads into the large LPG ship's girder. The numerical results of this study are pointing out that the non-lineariterative approach is necessary for the computation of the extreme loads induced by the oblique waves, ensuring better accuracy of the large LPG ship's longitudinal strength assessment.

A scheme for efficiently solving the nonlinear electromagnetic inverse scattering problem on sparse investigation domains is described. The proposed scheme reconstructs the (complex) dielectric permittivity of an investigation domain from fields measured away from the domain itself. Least-squares data misfit between the computed scattered fields, which are expressed as a nonlinear function of the permittivity, and the measured fields is constrained by the L0/L1-norm of the solution. The resulting minimization problem is solved using nonlinear Landweber iterations, where at each iteration a thresholding function is applied to enforce the sparseness-promoting L0/L1-norm constraint. The thresholded nonlinear Landweber iterations are applied to several two-dimensional problems, where the ``measured\\'\\' fields are synthetically generated or obtained from actual experiments. These numerical experiments demonstrate the accuracy, efficiency, and applicability of the proposed scheme in reconstructing sparse profiles with high permittivity values.

In spite of the tremendous progress achieved in recent years in the general area of iterative solution techniques, there are still a few obstacles to the acceptance of iterativemethods in a number of applications. These applications give rise to very indefinite or highly ill-conditioned non Hermitian matrices. Trying to solve these systems with the simple-minded standard preconditioned Krylov subspace methods can be a frustrating experience. With the mathematical and physical models becoming more sophisticated, the typical linear systems which we encounter today are far more difficult to solve than those of just a few years ago. This trend is likely to accentuate. This workshop will discuss (1) these applications and the types of problems that they give rise to; and (2) recent progress in solving these problems with iterativemethods. The workshop will end with a hopefully stimulating panel discussion with the speakers.

Some block iterativemethods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established.Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.

We present a general form of the iteration and interpolation process used in implicit particle filters. Implicit filters are based on a pseudo-Gaussian representation of posterior densities, and are designed to focus the particle paths so as to reduce the number of particles needed in nonlinear data assimilation. Examples are given.

In this paper we propose a modification of the Landweber iteration termed frozen Landweber iteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numerical performance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared with that of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based on the same convergence accuracy.

A method for rapid and accurate solution of linear and/or nonlinear thermal network problems is described. It is a matrix iterative process that converges for nodal temperatures and variations of thermal conductivity with temperature. The method is computer oriented and can be changed easily for design studies.

The development and optimization of medical ultrasound transducers and imaging modalities require a computational method that accurately predicts the nonlinear acoustic pressure field. A prospective method should provide the wide-angle, pulsed field emitted by an arbitrary planar source distribution

In this paper we discuss the construction, analysis and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg-Marquardt and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in the presence of data noise. Moreover, we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations.

Development and use of electromagnetic inverse scattering techniques for imagining sparse domains have been on the rise following the recent advancements in solving sparse optimization problems. Existing techniques rely on iteratively converting the nonlinear forward scattering operator into a sequence of linear ill-posed operations (for example using the Born iterativemethod) and applying sparsity constraints to the linear minimization problem of each iteration through the use of L0/L1-norm penalty term (A. Desmal and H. Bagci, IEEE Trans. Antennas Propag, 7, 3878–3884, 2014, and IEEE Trans. Geosci. Remote Sens., 3, 532–536, 2015). It has been shown that these techniques produce more accurate and sharper images than their counterparts which solve a minimization problem constrained with smoothness promoting L2-norm penalty term. But these existing techniques are only applicable to investigation domains involving weak scatterers because the linearization process breaks down for high values of dielectric permittivity.

A frequently faced task in experimental physics is to measure the probability distribution of some quantity. Often this quantity to be measured is smeared by a non-ideal detector response or by some physical process. The procedure of removing this smearing effect from the measured distribution is called unfolding, and is a delicate problem in signal processing. Due to the numerical ill-posedness of this task, various methods were invented which, given some assumptions on the initial probability distribution, try to regularize the problem. Most of these methods definitely introduce bias on the estimate of the initial probability distribution. We propose a linear iterativemethod (motivated by the Neumann series / Landweber iteration known in functional analysis), which has the advantage that no assumptions on the initial probability distribution is needed, and the only regularization parameter is the stopping order of the iteration. Convergence is proved under certain quite general conditions, which hold for p...

This paper deals with the monotone iterativemethod for set- valued operator equation in ordered normed space. Some results for the case of single valued operator are generalized here, as an application, a discontinuous nonlinear differential equation problem is discussed.

A Newton-BICGSTAB solver has been developed to reduce the CPU execution time of the FORMOSA-B boiling water reactor (BWR) core simulator. The new solver treats the strong non-linearities in the problem explicitly using the Newton's method, replacing the traditionally used nested iterative approach. Taking advantage of the higher convergence rate provided by the Newton's method, assuming that a good initial estimate of the unknowns is provided, and utilizing an efficient preconditioned BICGSTAB solver, we have developed a computationally efficient Newton-BICGSTAB solver to evaluate the three-dimensional, two-group neutron diffusion equations coupled with a two-phase flow model within a BWR core simulator. The robustness of the solver has been tested against numerous BWR core configurations and consistent results have been observed each time. The best exact Newton-BICGSTAB solver performance provides an overall speedup of 2.07 to the core simulator, with reference to the traditional approach, i.e. outer (fission-source)-inner (red/black line SOR). When solving the same problem using the traditional approach but with the BICGSTAB solver as the inner iteration solver [traditional (BICGSTAB)], we observed a speedup of 1.85. This means that the Newton-BICGSTAB solver provides an additional 12% increase in the overall speedup over the traditional (BICGSTAB) solver. However, one needs to note that, on average, the exact Newton-BICGSTAB solver provides an overall speedup of around 1.70; whereas, on average, the traditional (BICGSTAB) provides an overall speedup of around 1.60. An investigation on the feasibility of implementing an inexact Newton-BICGSTAB solver indicates that further reduction in the execution time can likely be obtained through this approach. This study shows that the inexact Newton-BICGSTAB solver can provide speedups of 1.73 to 2.10 with respect to the traditional solver.

Full Text Available In manifold learning, the aim of alignment is to derive the global coordinate of manifold from the local coordinates of manifold’s patches. At present, most of manifold learning algorithms assume that the relation between the global and local coordinates is locally linear and based on this linear relation align the local coordinates of manifold’s patches into the global coordinate of manifold. There are two contributions in this paper. First, the nonlinear relation between the manifold’s global and local coordinates is deduced by making use of the differentiation of local pullback functions defined on the differential manifold. Second, the method of local linear iterative alignment is used to align the manifold’s local coordinates into the manifold’s global coordinate. The experimental results presented in this paper show that the errors of noniterative alignment are considerably large and can be reduced to almost zero within the first two iterations. The large errors of noniterative/linear alignment verify the nonlinear nature of alignment and justify the necessity of iterative alignment.

By using partial order method, the existence, uniqueness and iterative ap-proximation of solutions for a class of systems of nonlinear operator equations in Banachspace are discussed. The results obtained in this paper extend and improve recent results.

We have developed and numerically evaluated an iterative algorithm for interferogram processing including the Fourier-transform method, the Gerchberg-Papoulis algorithm and Wiener's filter-based regularization used in combination. Using a signal-to-noise ratio not less than 1, it has been possible to reconstruct the phase of an object field with accuracy better than 5%.

This paper considers Newton-MSOR iterativemethod for solving 1D nonlinear porous medium equation (PME). The basic concept of proposed iterativemethod is derived from a combination of one step nonlineariterativemethod which known as Newton method with Modified Successive Over Relaxation (MSOR) method. The reliability of Newton-MSOR to obtain approximate solution for several PME problems is compared with Newton-Gauss-Seidel (Newton-GS) and Newton-Successive Over Relaxation (Newton-SOR). In this paper, the formulation and implementation of these three iterativemethods have also been presented. From four examples of PME problems, numerical results showed that Newton-MSOR method requires lesser number of iterations and computational time as compared with Newton-GS and Newton-SOR methods.

In the note we consider an iterative generalisation of the rejection sampling method. In high energy physics, this sampling is frequently used for event generation, i.e. preparation of phase space points distributed according to a matrix element squared $|M|^2$ for a scattering process. In many realistic cases $|M|^2$ is a complicated multi-dimensional function, so, the standard von Neumann procedure has quite low efficiency, even if an error reducing technique, like VEGAS, is applied. As a result of that, many of the $|M|^2$ calculations go to ``waste''. The considered iterative modification of the procedure can extract more ``unweighted'' events, i.e. distributed according to $|M|^2$. In several simple examples we show practical benefits of the technique and obtain more events than the standard von Neumann method, without any extra calculations of $|M|^2$.

This paper is concerned with a new discrete-time policy iteration adaptive dynamic programming (ADP) method for solving the infinite horizon optimal control problem of nonlinear systems. The idea is to use an iterative ADP technique to obtain the iterative control law, which optimizes the iterative performance index function. The main contribution of this paper is to analyze the convergence and stability properties of policy iterationmethod for discrete-time nonlinear systems for the first time. It shows that the iterative performance index function is nonincreasingly convergent to the optimal solution of the Hamilton-Jacobi-Bellman equation. It is also proven that any of the iterative control laws can stabilize the nonlinear systems. Neural networks are used to approximate the performance index function and compute the optimal control law, respectively, for facilitating the implementation of the iterative ADP algorithm, where the convergence of the weight matrices is analyzed. Finally, the numerical results and analysis are presented to illustrate the performance of the developed method.

The classical D-type iterative learning control law depends crucially on the relative degree of the controlled system, high order differential iterative learning law must be taken for systems with high order relative degree. It is very difficult to ascertain the relative degree of the controlled system for uncertain nonlinear systems. A first-order D-type iterative learning control design method is presented for a class of nonlinear systems with unknown relative degree based on dummy model in this paper. A dummy model with relative degree 1 is constructed for a class of nonlinear systems with unknown relative degree. A first-order D-type iterative learning control law is designed based on the dummy model, so that the dummy model can track the desired trajectory perfectly, and the controlled system can track the desired trajectory within a certain error. The simulation example demonstrates the feasibility and effectiveness of the presented method.

High nonlinearity is a typical characteristic associated with data assimilation systems. Additionally, iterative ensemble based methods have attracted a large amount of research attention, which has been focused on dealing with nonlinearity problems. To solve the local convergence problem of the iterative ensemble Kalman filter, a modified iterative ensemble Kalman filter algorithm was put forward, which was based on a global convergence strategy from the perspective of a Gauss-Newton iteration. Through self-adaption, the step factor was adjusted to enable every iteration to approach expected values during the process of the data assimilation. A sensitivity experiment was carried out in a low dimensional Lorenz-63 chaotic system, as well as a Lorenz-96 model. The new method was tested via ensemble size, observation variance, and inflation factor changes, along with other aspects. Meanwhile, comparative research was conducted with both a traditional ensemble Kalman filter and an iterative ensemble Kalman filter. The results showed that the modified iterative ensemble Kalman filter algorithm was a data assimilation method that was able to effectively estimate a strongly nonlinear system state.

Nonlinear functional analysis and applications is an area of study that has provided fascination for many mathematicians across the world. This monograph delves specifically into the topic of the geometric properties of Banach spaces and nonlineariterations, a subject of extensive research over the past thirty years. Chapters 1 to 5 develop materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained are summarized at the end of each chapter for easy reference. Chapters 6 to 23 deal with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings. This includes detailed workings on solutions of variational inequality problems, solutions of Hammerstein integral equations, and common fixed points (and common zeros) of families of nonlinear mappings. Carefully referenced and full of recent,...

Full Text Available We present another simple way of deriving several iterativemethods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newton's method. Also, we obtain well-known methods as special cases, for example, Halley's method, super-Halley method, Ostrowski's square-root method, Chebyshev's method, and so forth. Further, new classes of third-order multipoint iterativemethods free from a second-order derivative are derived by semidiscrete modifications of cubically convergent iterativemethods. Furthermore, a simple linear combination of two third-order multipoint iterativemethods is used for designing new optimal methods of order four.

In this paper, we use monotone iterative techniques to show the existence of maximal or minimal solutions of some elliptic PDEs with nonlinear discontinuous terms. As the numerical analysis of this PDEs is concerned, we prove the convergence of discrete extremal solutions.

In this paper, a value iteration adaptive dynamic programming (ADP) algorithm is developed to solve infinite horizon undiscounted optimal control problems for discrete-time nonlinear systems. The present value iteration ADP algorithm permits an arbitrary positive semi-definite function to initialize the algorithm. A novel convergence analysis is developed to guarantee that the iterative value function converges to the optimal performance index function. Initialized by different initial functions, it is proven that the iterative value function will be monotonically nonincreasing, monotonically nondecreasing, or nonmonotonic and will converge to the optimum. In this paper, for the first time, the admissibility properties of the iterative control laws are developed for value iteration algorithms. It is emphasized that new termination criteria are established to guarantee the effectiveness of the iterative control laws. Neural networks are used to approximate the iterative value function and compute the iterative control law, respectively, for facilitating the implementation of the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the present method.

This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regu-larization can quicken the convergence speed and reduce the calculation burden efficiently.

This paper proposes the new concurrent iterativemethods without using any derivatives for finding all zeros of polynomials simultaneously. The new methods are of monotonic convergence for both simple and multiple real-zeros of polynomials and are quadratically convergent. The corresponding accelerated concurrent iterativemethods are obtained too. The new methods are good candidates for the application in solving symmetric eigenproblems.

In this paper, we propose an iterative approach to increase the computation efficiency of the homotopy analysis method (HAM), a analytic technique for highly nonlinear problems. By means of the Schmidt-Gram process (Arfken et al., 1985) [15], we approximate the right-hand side terms of high-order linear sub-equations by a finite set of orthonormal bases. Based on this truncation technique, we introduce the Mth-order iterative HAM by using each Mth-order approximation as a new initial guess. It is found that the iterative HAM is much more efficient than the standard HAM without truncation, as illustrated by three nonlinear differential equations defined in an infinite domain as examples. This work might greatly improve the computational efficiency of the HAM and also the Mathematica package BVPh for nonlinear BVPs.

We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time s...

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are imnroved and generalized.

We develop a new iterativemethod for finding approximate solutions for spherical bounces associated with the decay of the false vacuum in scalar field theories. The method works for any generic potential in any number of dimensions, contains Coleman's thin-wall approximation as its first iteration, and greatly improves its accuracy by including higher order terms.

This paper points out that the so called enhanced variational iterationmethod (Colantoni & Boubaker, 2014) for a nonlinear equation arising in electrospinning and vibration-electrospinning process is the standard variational iterationmethod. An effective algorithm using the variational iteration algorithm-II is suggested for Bratu-like equation arising in electrospinning. A suitable choice of initial guess results in a relatively accurate solution by one or few iteration.

Control of the fusion power through regulation of the plasma density and temperature will be essential for achieving and maintaining desired operating points in fusion reactors and burning plasma experiments like ITER. In this work, a volume averaged model for the evolution of the density of energy, deuterium and tritium fuel ions, alpha-particles, and impurity ions is used to synthesize a multi-input multi-output nonlinear feedback controller for stabilizing and modulating the burn condition. Adaptive control techniques are used to account for uncertainty in model parameters, including particle confinement times and recycling rates. The control approach makes use of the different possible methods for altering the fusion power, including adjusting the temperature through auxiliary heating, modulating the density and isotopic mix through fueling, and altering the impurity density through impurity injection. Furthermore, a model-based optimization scheme is proposed to drive the system as close as possible to desired fusion power and temperature references. Constraints are considered in the optimization scheme to ensure that, for example, density and beta limits are avoided, and that optimal operation is achieved even when actuators reach saturation. Supported by the NSF CAREER award program (ECCS-0645086).

A weighted least-squares problem with a very ill-conditioned weight matrix arises in many applications. Because of round-off errors, the standard conjugate gradient method for solving this system does not give the correct answer even after n iterations. In this paper we propose an iterative algorithm based on a new type of reorthogonalization that converges to the solution.

Full Text Available The Variational IterationMethod (VIM and Modified Variational IterationMethod (MVIM are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problems were solved by Adomian Decomposition Method and VIM and Homotopy Perturbation Method. Comparisons with exact solutions reveal that the Variational IterationMethod (VIM and the Modified Variational IterationMethod (MVIM are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

Newton iterativemethods, also known as truncated Newton methods, are implementations of Newton`s method in which the linear systems that characterize Newton steps are solved approximately using iterative linear algebra methods. Here, we outline a well-developed Newton iterative algorithm together with a Fortran implementation called NITSOL. The basic algorithm is an inexact Newton method globalized by backtracking, in which each initial trial step is determined by applying an iterative linear solver until an inexact Newton criterion is satisfied. In the implementation, the user can specify inexact Newton criteria in several ways and select an iterative linear solver from among several popular {open_quotes}transpose-free{close_quotes} Krylov subspace methods. Jacobian-vector products used by the Krylov solver can be either evaluated analytically with a user-supplied routine or approximated using finite differences of function values. A flexible interface permits a wide variety of preconditioning strategies and allows the user to define a preconditioner and optionally update it periodically. We give details of these and other features and demonstrate the performance of the implementation on a representative set of test problems.

This paper presents a new iterative technique for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions. The method is based on the homotopy perturbation method and the integral equation formalism in which a recursive scheme is established for the components of the approximate series solution. This method does not involve solution of a sequence of nonlinear algebraic or transcendental equations for the unknown coefficients as in some other iterative techniques developed for singular boundary value problems. The convergence result for the proposed method is established in the paper. The method is illustrated by four numerical examples, two of which have physical significance: The first problem is an application of the reaction-diffusion process in a porous spherical catalyst and the second problem arises in the study of steady-state oxygen-diffusion in a spherical cell with Michaelis-Menten uptake kinetics.

It is well known that many iterativemethods fail when applied to nonlinear systems of convection-dominated transport equations. Most successful methods for obtaining steady-state solutions to such systems rely on time-stepping through an artificial transient, combined with careful construction of artificial dissipation operators. These operators provide control over spurious oscillations which pollute the steady state solutions, and, in the nonlinear case, may become amplified and lead to instability. In the present study, we investigate Taylor Galerkin and SUPG-type methods and compare results for steady-state solutions to the Euler equations of gas dynamics. In particular, we consider the efficiency of different iterative strategies and present results for representative two-dimensional calculations.

Self-focusing is observed in nonlinear materials owing to the interaction between laser and matter when laser beam propagates. Some of numerical simulation strategies such as the beam propagation method (BPM) based on nonlinear Schrödinger equation and ray tracing method based on Fermat's principle have applied to simulate the self-focusing process. In this paper we present an iterationnonlinear ray tracing method in that the nonlinear material is also cut into massive slices just like the existing approaches, but instead of paraxial approximation and split-step Fourier transform, a large quantity of sampled real rays are traced step by step through the system with changing refractive index and laser intensity by iteration. In this process a smooth treatment is employed to generate a laser density distribution at each slice to decrease the error caused by the under-sampling. The characteristics of this method is that the nonlinear refractive indices of the points on current slice are calculated by iteration so as to solve the problem of unknown parameters in the material caused by the causal relationship between laser intensity and nonlinear refractive index. Compared with the beam propagation method, this algorithm is more suitable for engineering application with lower time complexity, and has the calculation capacity for numerical simulation of self-focusing process in the systems including both of linear and nonlinear optical media. If the sampled rays are traced with their complex amplitudes and light paths or phases, it will be possible to simulate the superposition effects of different beam. At the end of the paper, the advantages and disadvantages of this algorithm are discussed.

In this paper we apply methods of proof mining to obtain a highly uniform effective rate of asymptotic regularity for the Ishikawa iteration associated to nonexpansive self-mappings of convex subsets of a class of uniformly convex geodesic spaces. Moreover, we show that these results are guaranteed by a combination of logical metatheorems for classical and semi-intuitionistic systems.

Full Text Available The dynamics of a biparametric family for solving nonlinear equations is studied on quadratic polynomials. This biparametric family includes the c-iterativemethods and the well-known Chebyshev-Halley family. We find the analytical expressions for the fixed and critical points by solving 6-degree polynomials. We use the free critical points to get the parameter planes and, by observing them, we specify some values of (α, c with clear stable and unstable behaviors.

对于一类非线性信号的去噪问题,该文提出一种基于奇异值分解(Singular Value Decomposition, SVD)的有效迭代方法.对现有奇异值差分谱方法在两类不同非线性信号上的去噪效果进行了对比,指出在信号不具有明显特征频率、非周期性变化时这一方法并不适用,并分析了现象产生的原因;然后针对该类信号的特点重新定义了Hankel矩阵结构,给出有效奇异值的确定方式,并通过SVD多次迭代过程实现对该类信号的有效去噪.对实际飞行数据去噪的实验结果表明,该方法对提出的一类信号对象不仅去噪效果良好,而且可提高运算效率.%To solve a class of nonlinear signal denoising, an effective iterationmethod based on the Singular Value Decomposition (SVD) is proposed. When the signals have no obvious characteristic frequency and non-periodic change, the current difference spectrum method is not applicable by comparing the results on the two class of nonlinear signal, and then the corresponding reason is analyzed. According to the signal feature, the structure of the Hankel matrix is defined again and the valid singular values are determined. The effective denoising is realized by the repeated iteration which is based on the SVD. The results of the flight data demonstrate that the proposed method can effectively reduce the noise and improve the computing efficiency as well.

This paper presents a nonlinear model-based iterative learning control procedure to achieve accurate tracking control for nonlinear lumped mechanical continuous-time systems. The model structure used in this iterative learning control procedure is new and combines a linear state space model and a nonlinear feature space transformation. An intuitive two-step iterative algorithm to identify the model parameters is presented. It alternates between the estimation of the linear and the nonlinear model part. It is assumed that besides the input and output signals also the full state vector of the system is available for identification. A measurement and signal processing procedure to estimate these signals for lumped mechanical systems is presented. The iterative learning control procedure relies on the calculation of the input that generates a given model output, so-called offline model inversion. A new offline nonlinear model inversion method for continuous-time, nonlinear time-invariant, state space models based on Newton's method is presented and applied to the new model structure. This model inversion method is not restricted to minimum phase models. It requires only calculation of the first order derivatives of the state space model and is applicable to multivariable models. For periodic reference signals the method yields a compact implementation in the frequency domain. Moreover it is shown that a bandwidth can be specified up to which learning is allowed when using this inversion method in the iterative learning control procedure. Experimental results for a nonlinear single-input-single-output system corresponding to a quarter car on a hydraulic test rig are presented. It is shown that the new nonlinear approach outperforms the linear iterative learning control approach which is currently used in the automotive industry on durability test rigs.

A new algebraic reconstruction technique (ART), nonlinear auto-adjusting iterative reconstruction technique (NAIRT), is proposed and applied to reconstruct a section of an actual thermal air flow field. With numerical simulation, NAIRT was tested to reconstruct a complicated field to demonstrate its superior reconstructive capability. In contrast, three typical ARTs, the basic ART, simultaneous ART (SART), and a modified SART (MSART), were simulated to demonstrate the reconstructive capability improvement attained through the use of the proposed NAIRT. The calculated results were discussed with mean square error (MSE) and peak error (PE). A thermal air flow field was produced with an alcohol burner and was detected by a laser beam. With laser beam projections, a cross-section of the field was reconstructed by NAIRT. As a result, the reconstructive capability was improved much by NAIRT. The MSE decreased by 95.5%, and PE by 97.2% from that of the basic ART. Only NAIRT converged without filters while its reconstructive accuracy improved. By increasing the projections from 42 to 84, the accuracy of NAIRT without filters was improved significantly. NAIRT could effectively reconstruct the section of the thermal field. The proposed NAIRT needed no filter for its convergence and it had the highest reconstructive accuracy and simplest iterative expression of those analyzed.

We study the application of enhanced nonlineariterativemethods to the steady-state solution of a system of two-dimensional convection-diffusion-reaction partial differential equations that describe the partially-ionized plasma flow in the boundary layer of a tokamak fusion reactor. This system of equations is characterized by multiple time and spatial scales, and contains highly anisotropic transport coefficients due to a strong imposed magnetic field. We use Newton`s method to linearize the nonlinear system of equations resulting from an implicit, finite volume discretization of the governing partial differential equations, on a staggered Cartesian mesh. The resulting linear systems are neither symmetric nor positive definite, and are poorly conditioned. Preconditioned Krylov iterative techniques are employed to solve these linear systems. We investigate both a modified and a matrix-free Newton-Krylov implementation, with the goal of reducing CPU cost associated with the numerical formation of the Jacobian. A combination of a damped iteration, one-way multigrid and a pseudo-transient continuation technique are used to enhance global nonlinear convergence and CPU efficiency. GMRES is employed as the Krylov method with Incomplete Lower-Upper(ILU) factorization preconditioning. The goal is to construct a combination of nonlinear and linear iterative techniques for this complex physical problem that optimizes trade-offs between robustness, CPU time, memory requirements, and code complexity. It is shown that a one-way multigrid implementation provides significant CPU savings for fine grid calculations. Performance comparisons of the modified Newton-Krylov and matrix-free Newton-Krylov algorithms will be presented.

We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation.The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iterationmethod.

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iterationmethod, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...

We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time steps, than what is customary in the Brinkman penalization, thus reducing its computational cost while maintaining the capability of the method to handle complex geometries. We demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate. We find that the present method enhances significantly the accuracy of the Brinkman penalization technique for the simulations of highly unsteady flows past complex geometries.

Full Text Available Interferometry filtering is one of the key steps in obtain high-precision Digital Elevation Model (DEM and Digital Orthophoto Map (DOM. In the case of low-correlation or complicated topography, traditional phase filtering methods fail in balancing noise elimination and phase preservation, which leads to inaccurate interferometric phase. This paper proposed an adaptive iterated nonlocal interferometry filtering method to deal with the problem. Based on the thought of nonlocal filtering, the proposed method filters the image with utilization of the image redundancy information. The smoothing parameter of the method is adaptive to the interferometry, and automatic iteration, in which the window size is adjusted, is applied to improve the filtering precision. Validity of the proposed method is verified by simulated and real data. Comparison with existed methods is given at the same time.

For the electro-hydraulic servo vibrating system(ESVS) with the characteristics of non-linearity and repeating motion, a novel method, PI-type iterative learning control(ILC), is proposed on the basis of traditional PID control. By using memory ability of computer, the method keeps last time's tracking error of the system and then applies the error information to the next time's control process. At the same time, a forgetting factor and a D-type learning law of feedforward fuzzy-inferring referenced displacement error under the optimal objective are employed to enhance the systemic robustness and tracking accuracy. The results of simulation and test reveal that the algorithm has a trait of high repeating precision, and could restrain the influence of nonlinear factors like leaking, external disturbance, aerated oil, etc. Compared with traditional PID control, it could better meet the requirement of nonlinear electro-hydraulic servo vibrating system.

Full Text Available Aggregation-iterative algorithms for linear operator equations are constructed and investigated. These algorithms cover methods of iterative aggregation and projection-iterativemethods. In convergence conditions there is neither requirement for the corresponding operator of fixed sign no restriction to the spectral radius to be less than one.

Full Text Available We define the concept of the generalized -concave operators, which generalize the definition of the -concave operators. By using the iterativemethod and the partial ordering method, we prove the existence and uniqueness of fixed points of this class of the operators. As an example of the application of our results, we show the existence and uniqueness of solutions to a class of the Hammerstein integral equations.

In this paper, an online identification algorithm is presented for nonlinear systems in the presence of output colored noise. The proposed method is based on extended recursive least squares (ERLS) algorithm, where the identified system is in polynomial Wiener form. To this end, an unknown intermediate signal is estimated by using an inner iterative algorithm. The iterative recursive algorithm adaptively modifies the vector of parameters of the presented Wiener model when the system parameters vary. In addition, to increase the robustness of the proposed method against variations, a robust RLS algorithm is applied to the model. Simulation results are provided to show the effectiveness of the proposed approach. Results confirm that the proposed method has fast convergence rate with robust characteristics, which increases the efficiency of the proposed model and identification approach. For instance, the FIT criterion will be achieved 92% in CSTR process where about 400 data is used.

Full Text Available We employed different iterationmethods like Homotopy Analysis Method (HAM, Adomian Decomposition Method (ADM, and Variational IterationMethod (VIM to find the approximate solution to the Zabolotskaya-Khokhlov (ZK equation. Iterationmethods are used to solve linear and nonlinear PDEs whose classical methods are either very complex or too limited to apply. A comparison study has been made to see which of these methods converges to the approximate solution rapidly. The result revealed that, amongst these methods, ADM is more effective and simpler tool in its nature which does not require any transformation or linearization.

An observer-based adaptive iterative learning control (AILC) scheme is developed for a class of nonlinear systems with unknown time-varying parameters and unknown time-varying delays. The linear matrix inequality (LMI) method is employed to design the nonlinear observer. The designed controller contains a proportional-integral-derivative (PID) feedback term in time domain. The learning law of unknown constant parameter is differential-difference-type, and the learning law of unknown time-varying parameter is difference-type. It is assumed that the unknown delay-dependent uncertainty is nonlinearly parameterized. By constructing a Lyapunov-Krasovskii-like composite energy function (CEF), we prove the boundedness of all closed-loop signals and the convergence of tracking error. A simulation example is provided to illustrate the effectiveness of the control algorithm proposed in this paper.

The connection between the solution of linear systems of equations by iterativemethods and explicit time stepping techniques is used to accelerate to steady state the solution of ODE systems arising from discretized PDEs which may involve either physical or artificial transient terms. Specifically, a class of Runge-Kutta (RK) time integration schemes with extended stability domains has been used to develop recursion formulas which lead to accelerated iterative performance. The coefficients for the RK schemes are chosen based on the theory of Chebyshev iteration polynomials in conjunction with a local linear stability analysis. We refer to these schemes as Chebyshev Parameterized Runge Kutta (CPRK) methods. CPRK methods of one to four stages are derived as functions of the parameters which describe an ellipse {Epsilon} which the stability domain of the methods is known to contain. Of particular interest are two-stage, first-order CPRK and four-stage, first-order methods. It is found that the former method can be identified with any two-stage RK method through the correct choice of parameters. The latter method is found to have a wide range of stability domains, with a maximum extension of 32 along the real axis. Recursion performance results are presented below for a model linear convection-diffusion problem as well as non-linear fluid flow problems discretized by both finite-difference and finite-element methods.

We study the regularization properties of iterative minimum-residual methods applied to discrete ill-posed problems. In these methods, the projection onto the underlying Krylov subspace acts as a regularizer, and the emphasis of this work is on the role played by the basis vectors of these Krylov...... subspaces. We provide a combination of theory and numerical examples, and our analysis confirms the experience that MINRES and MR-II can work as general regularization methods. We also demonstrate theoretically and experimentally that the same is not true, in general, for GMRES and RRGMRES their success...... as regularization methods is highly problem dependent....

We study the regularization properties of iterative minimum-residual methods applied to discrete ill-posed problems. In these methods, the projection onto the underlying Krylov subspace acts as a regularizer, and the emphasis of this work is on the role played by the basis vectors of these Krylov...... subspaces. We provide a combination of theory and numerical examples, and our analysis confirms the experience that MINRES and MR-II can work as general regularization methods. We also demonstrate theoretically and experimentally that the same is not true, in general, for GMRES and RRGMRES - their success...... as regularization methods is highly problem dependent....

We study the regularization properties of iterative minimum-residual methods applied to discrete ill-posed problems. In these methods, the projection onto the underlying Krylov subspace acts as a regularizer, and the emphasis of this work is on the role played by the basis vectors of these Krylov...... subspaces. We provide a combination of theory and numerical examples, and our analysis confirms the experience that MINRES and MR-II can work as general regularization methods. We also demonstrate theoretically and experimentally that the same is not true, in general, for GMRES and RRGMRES their success...

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterativemethods are considered. The implicit iterativemethod is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.

In the present paper, the new iterativemethod proposed by Daftardar-Gejji and Jafari (NIM or DJM) (2006) is used to solve the linear and nonlinear Fokker-Planck equations and some similar equations. In this iterativemethod the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian decomposition method (ADM). It does not require to calculate Lagrange multiplier as in variational iterationmethod (VIM) and for solving a nonlinear case, the terms of the sequence become complex after several iterations. Thus, analytical evaluation of terms becomes very difficult or impossible in VIM. No needs to construct a homotopy and solve the corresponding algebraic equations as in homotopy perturbation method (HPM). In this work, the applications of the DJM for 1D, 2D, 3D linear and nonlinear Fokker-Planck equations are given and the results demonstrate that the presented method is very effective and reliable and does not require any restrictive assumptions for nonlinear terms and provide the analytic solutions. A symbolic manipulator Mathematica® 10.0 was used to evaluate terms in the iterative process.

In this paper, we present a numerical scheme for the analysis of steady-state nonlinear diffusion in random heterogeneous media. The key idea is to iteratively solve the nonlinear stochastic governing equations via an inexact Picard iteration scheme, wherein the nonlinear constitutive law is linearized using the current guess of the solution. The linearized stochastic governing equations are then spatially discretized and approximately solved using stochastic reduced basis projection schemes. The approximation to the solution process thus obtained is used as the guess for the next iteration. This iterative procedure is repeated until an appropriate convergence criterion is met. Detailed numerical studies are presented for diffusion in a square domain for varying degrees of nonlinearity. The numerical results are compared against benchmark Monte Carlo simulations, and it is shown that the proposed approach provides good approximations for the response statistics at modest computational effort.

Full Text Available This paper focuses on the parameter identification problem for Wiener nonlinear dynamic systems with moving average noises. In order to improve the convergence rate, the gradient-based iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates, and to compute iteratively the noise estimates based on the obtained parameter estimates. The simulation results show that the proposed algorithm can effectively estimate the parameters of Wiener systems with moving average noises.

This Letter compares the classical variational iterationmethod with the fractional variational iterationmethod. The fractional complex transform is introduced to convert a fractional differential equation to its differential partner, so that its variational iteration algorithm can be simply constructed. -- Highlights: → The variational iterationmethod and its fractional modification are compared. → The demerits arising are overcome by the fractional complex transform. → The Letter provides a powerful tool to solving fractional differential equations.

H∞ control is a powerful method to solve the disturbance attenuation problems that occur in some control systems. The design of such controllers relies on solving the zero-sum game (ZSG). But in practical applications, the exact dynamics is mostly unknown. Identification of dynamics also produces errors that are detrimental to the control performance. To overcome this problem, an iterative adaptive dynamic programming algorithm is proposed in this paper to solve the continuous-time, unknown nonlinear ZSG with only online data. A model-free approach to the Hamilton-Jacobi-Isaacs equation is developed based on the policy iterationmethod. Control and disturbance policies and value are approximated by neural networks (NNs) under the critic-actor-disturber structure. The NN weights are solved by the least-squares method. According to the theoretical analysis, our algorithm is equivalent to a Gauss-Newton method solving an optimization problem, and it converges uniformly to the optimal solution. The online data can also be used repeatedly, which is highly efficient. Simulation results demonstrate its feasibility to solve the unknown nonlinear ZSG. When compared with other algorithms, it saves a significant amount of online measurement time.

This paper focuses on the search for suitable research methods for research regarding iteration in instructional design. More specifically my research concerned the question how instructional designers can be supported during an iterative design process. Although instructional design and development

The iterative technique of sign-changing solution is studied for a nonlinear third-order two-point boundary value problem, where the nonlinear term has the time sin-gularity. By applying the monotonically iterative technique, an existence theorem is established and two useful iterative schemes are obtained.

The analysis and prediction of time series data have been addressed by researchers. Many techniques have been developed to be applied in various areas, such as weather forecasting, financial markets and hydrological phenomena involving data that are contaminated by noise. Therefore, various techniques to improve the method have been introduced to analyze and predict time series data. In respect of the importance of analysis and the accuracy of the prediction result, a study was undertaken to test the effectiveness of the improved nonlinear prediction method for data that contain noise. The improved nonlinear prediction method involves the formation of composite serial data based on the successive differences of the time series. Then, the phase space reconstruction was performed on the composite data (one-dimensional) to reconstruct a number of space dimensions. Finally the local linear approximation method was employed to make a prediction based on the phase space. This improved method was tested with data series Logistics that contain 0%, 5%, 10%, 20% and 30% of noise. The results show that by using the improved method, the predictions were found to be in close agreement with the observed ones. The correlation coefficient was close to one when the improved method was applied on data with up to 10% noise. Thus, an improvement to analyze data with noise without involving any noise reduction method was introduced to predict the time series data.

We consider several synchronous and asynchronous multisplitting iteration schemes for solving a class of nonlinear complementarity problems with the system matrix being an H-matrix. We establish the convergence theorems for the schemes. The numerical experiments show that the schemes are efficient for solving the class of nonlinear complementarity problems.

An estimation method for solutions to the general linear system of Volterratype integral inequalities containing several iterated integral functionals is obtained. This method is based on a result proved by the present second author in Journ. Math. Anal. Appl.(1984). A certain two-dimensional system of nonlinear ordinary differential equations is also discussed to demonstrate the usefulness of our method.

In this paper the author will give a survey on iterativemethods for solving linear equations with Toeplitz matrices, Block Toeplitz matrices, Toeplitz plus Hankel matrices, and matrices with low displacement rank. He will treat the following subjects: (1) optimal (w)-circulant preconditioners is a generalization of circulant preconditioners; (2) Optimal implementation of circulant-like preconditioners in the complex and real case; (3) preconditioning of near-singular matrices; what kind of preconditioners can be used in this case; (4) circulant preconditioning for more general classes of Toeplitz matrices; what can be said about matrices with coefficients that are not l{sub 1}-sequences; (5) preconditioners for Toeplitz least squares problems, for block Toeplitz matrices, and for Toeplitz plus Hankel matrices.

Data coming from different sources have different types and temporal states. Relations between one type of data and another ones, or between data and unknown parameters are almost nonlinear. It is not accurate and reliable to process the data in building the digital earth with the classical least squares method or the method of the common nonlinear least squares. So a generalized nonlinear dynamic least squares method was put forward to process data in building the digital earth. A separating solution model and the iterative calculation method were used to solve the generalized nonlinear dynamic least squares problem. In fact, a complex problem can be separated and then solved by converting to two sub-problems, each of which has a single variable. Therefore the dimension of unknown parameters can be reduced to its half, which simplifies the original high dimensional equations.

This paper concerns with the statistical methods for solving general linear systems.After a brief review of Bayesian perspective for inverse problems,a new and efficient iterativemethod for general linear systems from a Bayesian perspective is proposed.The convergence of this iterativemethod is proved,and the corresponding error analysis is studied.Finally,numerical experiments are given to support the efficiency of this iterativemethod,and some conclusions are obtained.

Full Text Available Let K be a nonempty, bounded, closed, and convex subset of a Banach space. We show that the iterates of a typical element (in the sense of Baire's categories of a class of continuous self-mappings of K converge uniformly on K to the unique fixed point of this typical element.

The general mixed quasi variational inequality containing a nonlinear term ψ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term ψ are equivalent to the fixed point problems and re, solvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterativemethod for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.

Full Text Available We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterativemethod as discretization of the regularization method. We prove that both regularization and iterativemethods converge in norm.

We study the iteration of block spin transformations in the O(3) symmetric nonlinear {sigma} model on a two-dimensional square lattice with the help of the Monte Carlo method. In contrast with the classical Monte Carlo renormalization group approach, we {ital do} attempt to explicitly compute the block spin effective actions. Using two different methods for the determination of effective couplings, we study the renormalization group flow for various parametrization and truncation schemes. The largest ansatz for the effective action contains thirteen coupling constants. Actions on the renormalized trajectory should describe theories with no lattice artifacts, even at a small correlation length. However, tests with the step scaling function of L{umlt u}scher, Weisz, and Wolff reveal that our truncated effective actions show sizable scaling violations indicating that the {ital Ans{umlt a}tze} are still too small. {copyright} {ital 1996 The American Physical Society.}

Since their early application to elliptic partial differential equations, multigrid methods have been applied successfully to a large and growing class of problems, from elasticity and computational fluid dynamics to geodetics and molecular structures. Classical multigrid begins with a two-grid process. First, iterative relaxation is applied, whose effect is to smooth the error. Then a coarse-grid correction is applied, in which the smooth error is determined on a coarser grid. This error is interpolated to the fine grid and used to correct the fine-grid approximation. Applying this method recursively to solve the coarse-grid problem leads to multigrid. The coarse-grid correction works because the residual equation is linear. But this is not the case for nonlinear problems, and different strategies must be employed. In this presentation we describe how to apply multigrid to nonlinear problems. There are two basic approaches. The first is to apply a linearization scheme, such as the Newton's method, and to employ multigrid for the solution of the Jacobian system in each iteration. The second is to apply multigrid directly to the nonlinear problem by employing the so-called Full Approximation Scheme (FAS). In FAS a nonlineariteration is applied to smooth the error. The full equation is solved on the coarse grid, after which the coarse-grid error is extracted from the solution. This correction is then interpolated and applied to the fine grid approximation. We describe these methods in detail, and present numerical experiments that indicate the efficacy of them.

We investigate possibilities to speed up iterative algorithms for non-blind image deconvolution. We focus on algorithms in which convolution with the point-spread function to be deconvolved is used in each iteration, and aim at accelerating these convolution operations as they are typically the most expensive part of the computation. We follow two approaches: First, for some practically important specific point-spread functions, algorithmically efficient sliding window or list processing tech...

This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time.

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger's equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterativemethod is easy to implement and accurate when applied to time-fractional partial differential equations.

We present an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in vortex particle methods. This is achieved by implementing a penalization of the velocity field using iteration of the penalized vorticity. We show that using the conventional Brinkman pen...

In this paper we examine the dramatic influence that a severe stretching of finite difference grids can have on the convergence behaviour of iterativemethods. For the most important classes of iterativemethods this phenomenon is considered for a simple model problem with various boundary

In this paper, an adaptive iterative learning control scheme is proposed for a class of non-linearly parameterised systems with unknown time-varying parameters and input saturations. By incorporating a saturation function, a new iterative learning control mechanism is presented which includes a feedback term and a parameter updating term. Through the use of parameter separation technique, the non-linear parameters are separated from the non-linear function and then a saturated difference updating law is designed in iteration domain by combining the unknown parametric term of the local Lipschitz continuous function and the unknown time-varying gain into an unknown time-varying function. The analysis of convergence is based on a time-weighted Lyapunov-Krasovskii-like composite energy function which consists of time-weighted input, state and parameter estimation information. The proposed learning control mechanism warrants a L2[0, T] convergence of the tracking error sequence along the iteration axis. Simulation results are provided to illustrate the effectiveness of the adaptive iterative learning control scheme.

A new filter named the maximum likelihood-based iterated divided difference filter (MLIDDF) is developed to improve the low state estimation accuracy of nonlinear state estimation due to large initial estimation errors and nonlinearity of measurement equations. The MLIDDF algorithm is derivative-free and implemented only by calculating the functional evaluations. The MLIDDF algorithm involves the use of the iteration measurement update and the current measurement, and the iteration termination criterion based on maximum likelihood is introduced in the measurement update step, so the MLIDDF is guaranteed to produce a sequence estimate that moves up the maximum likelihood surface. In a simulation, its performance is compared against that of the unscented Kalman filter (UKF), divided difference filter (DDF), iterated unscented Kalman filter (IUKF) and iterated divided difference filter (IDDF) both using a traditional iteration strategy. Simulation results demonstrate that the accumulated mean-square root error for the MLIDDF algorithm in position is reduced by 63% compared to that of UKF and DDF algorithms, and by 7% compared to that of IUKF and IDDF algorithms. The new algorithm thus has better state estimation accuracy and a fast convergence rate.

Full Text Available A new filter named the maximum likelihood-based iterated divided difference filter (MLIDDF is developed to improve the low state estimation accuracy of nonlinear state estimation due to large initial estimation errors and nonlinearity of measurement equations. The MLIDDF algorithm is derivative-free and implemented only by calculating the functional evaluations. The MLIDDF algorithm involves the use of the iteration measurement update and the current measurement, and the iteration termination criterion based on maximum likelihood is introduced in the measurement update step, so the MLIDDF is guaranteed to produce a sequence estimate that moves up the maximum likelihood surface. In a simulation, its performance is compared against that of the unscented Kalman filter (UKF, divided difference filter (DDF, iterated unscented Kalman filter (IUKF and iterated divided difference filter (IDDF both using a traditional iteration strategy. Simulation results demonstrate that the accumulated mean-square root error for the MLIDDF algorithm in position is reduced by 63% compared to that of UKF and DDF algorithms, and by 7% compared to that of IUKF and IDDF algorithms. The new algorithm thus has better state estimation accuracy and a fast convergence rate.

Full Text Available Iterative solution methods to solve linear systems of equations were originally formulated as basic iterationmethods of defect-correction type, commonly referred to as Richardson's iterationmethod. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iterationmethods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.

Two iterativemethods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.

Full Text Available A model for dynamic analysis of the vehicle-track nonlinear coupling system is established by the finite element method. The whole system is divided into two subsystems: the vehicle subsystem and the track subsystem. Coupling of the two subsystems is achieved by equilibrium conditions for wheel-to-rail nonlinear contact forces and geometrical compatibility conditions. To solve the nonlinear dynamics equations for the vehicle-track coupling system, a cross iteration algorithm and a relaxation technique are presented. Examples of vibration analysis of the vehicle and slab track coupling system induced by China’s high speed train CRH3 are given. In the computation, the influences of linear and nonlinear wheel-to-rail contact models and different train speeds are considered. It is found that the cross iteration algorithm and the relaxation technique have the following advantages: simple programming; fast convergence; shorter computation time; and greater accuracy. The analyzed dynamic responses for the vehicle and the track with the wheel-to-rail linear contact model are greater than those with the wheel-to-rail nonlinear contact model, where the increasing range of the displacement and the acceleration is about 10%, and the increasing range of the wheel-to-rail contact force is less than 5%.

Full Text Available A nonlinear detection technique designed for multiple-antenna assisted receivers employed in space-division multiple-access systems is investigated. We derive the optimal solution of the nonlinear spatial-processing assisted receiver for binary phase shift keying signalling, which we refer to as the Bayesian detector. It is shown that this optimal Bayesian receiver significantly outperforms the standard linear beamforming assisted receiver in terms of a reduced bit error rate, at the expense of an increased complexity, while the achievable system capacity is substantially enhanced with the advent of employing nonlinear detection. Specifically, when the spatial separation expressed in terms of the angle of arrival between the desired and interfering signals is below a certain threshold, a linear beamformer would fail to separate them, while a nonlinear detection assisted receiver is still capable of performing adequately. The adaptive implementation of the optimal Bayesian detector can be realized using a radial basis function network. Two techniques are presented for constructing block-data-based adaptive nonlinear multiple-antenna assisted receivers. One of them is based on the relevance vector machine invoked for classification, while the other on the orthogonal forward selection procedure combined with the Fisher ratio class-separability measure. A recursive sample-by-sample adaptation procedure is also proposed for training nonlinear detectors based on an amalgam of enhanced -means clustering techniques and the recursive least squares algorithm.

Commonly used time domain equalizer (TEQ) design methods have been recently unified as an optimization problem involving an objective function in the form of a Rayleigh quotient. The direct generalized eigenvalue solution relies on matrix decompositions. To reduce implementation complexity, we propose an iterative refinement approach in which the TEQ length starts at two taps and increases by one tap at each iteration. Each iteration involves matrix-vector multiplications and vector additions with[InlineEquation not available: see fulltext.] matrices and two-element vectors. At each iteration, the optimization of the objective function either improves or the approach terminates. The iterative refinement approach provides a range of communication performance versus implementation complexity tradeoffs for any TEQ method that fits the Rayleigh quotient framework. We apply the proposed approach to three such TEQ design methods: maximum shortening signal-to-noise ratio, minimum intersymbol interference, and minimum delay spread.

Full Text Available Commonly used time domain equalizer (TEQ design methods have been recently unified as an optimization problem involving an objective function in the form of a Rayleigh quotient. The direct generalized eigenvalue solution relies on matrix decompositions. To reduce implementation complexity, we propose an iterative refinement approach in which the TEQ length starts at two taps and increases by one tap at each iteration. Each iteration involves matrix-vector multiplications and vector additions with matrices and two-element vectors. At each iteration, the optimization of the objective function either improves or the approach terminates. The iterative refinement approach provides a range of communication performance versus implementation complexity tradeoffs for any TEQ method that fits the Rayleigh quotient framework. We apply the proposed approach to three such TEQ design methods: maximum shortening signal-to-noise ratio, minimum intersymbol interference, and minimum delay spread.

In this paper we develop multilevel iterationmethods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.

A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.

A robust self-calibration method is presented, which can efficiently discard the outliers based on a Weighted IterationMethod (WIM). The method is an iterative process in which the projective reconstruction is obtained based on the weights of all the points, whereas the weights are defined in inverse proportion to the reciprocal of the re-projective errors. The weights of outliers trend to zero after several iterations, and the accurate projective reconstruction is determined. The location of the absolute conic and the camera intrinsic parameters are obtained after the projective reconstruction. The theory and experiments with both simulate and real data demonstrate that the proposed method is very efficient and robust.

Full Text Available We apply the variational iterationmethod using He's polynomials (VIMHP for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

ITER is an international experimental facility being built by seven Parties to demonstrate the long term potential of fusion energy. The ITER Joint Implementation Agreement (JIA) defines the structure and governance model of such cooperation. There are a number of necessary conditions for such international projects to be successful: a complete design, strong systems engineering working with an agreed set of requirements, an experienced organization with systems and plans in place to manage the project, a cost estimate backed by industry, and someone in charge. Unfortunately for ITER many of these conditions were not present. The paper discusses the priorities in the JIA which led to setting up the project with a Central Integrating Organization (IO) in Cadarache, France as the ITER HQ, and seven Domestic Agencies (DAs) located in the countries of the Parties, responsible for delivering 90%+ of the project hardware as Contributions-in-Kind and also financial contributions to the IO, as ``Contributions-in-Cash.'' Theoretically the Director General (DG) is responsible for everything. In practice the DG does not have the power to control the work of the DAs, and there is not an effective management structure enabling the IO and the DAs to arbitrate disputes, so the project is not really managed, but is a loose collaboration of competing interests. Any DA can effectively block a decision reached by the DG. Inefficiencies in completing design while setting up a competent organization from scratch contributed to the delays and cost increases during the initial few years. So did the fact that the original estimate was not developed from industry input. Unforeseen inflation and market demand on certain commodities/materials further exacerbated the cost increases. Since then, improvements are debatable. Does this mean that the governance model of ITER is a wrong model for international scientific cooperation? I do not believe so. Had the necessary conditions for success

A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterativemethods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterativemethods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

In this Letter, a mathematical model of the problem of prey and predator is presented and He's variational iterationmethod is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The results are compared with the results obtained by Adomian decomposition method and homotopy perturbation method. Comparison of the methods show that He's variational iterationmethod is a powerful method for obtaining approximate solutions to nonlinear equations and their systems.

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He's variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iterationmethod (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303

Algebraic iterativemethods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely...... a fixed relaxation parameter in each method, namely, the one that leads to the fastest semiconvergence. Computational results show that for multicore computers, the sequential approach is preferable....

The preconditioned methods for solving linear system are discussed The convergence rate of accelerated overrelaxation (AOR) method can be enlarged by using the preconditioned method when the classical AOR method converges, and the preconditioned method is invalid when the classical iterativemethod does not converge. The results in corresponding references are improved and perfected.

In 1997, Kohno et al. have reported numerically that the improving modified Gauss-Seidel method, which was referred to as the IMGS method, is superior to the SOR iterativemethod. In this paper, we prove that the spectral radius of the IMGS method is smaller than that of the SOR method and Gauss-Seidel method, if the relaxation parameter [omega][set membership, variant](0,1]. As a result, we prove theoretically that this method is succeeded in improving the convergence of some classical iterativemethods. Some recent results are improved.

Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multigrid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction.

There are two main types of NRBCs: approximate local NRBCs and exact nonlocal ... For characterizing the behaviors of geologic materials, a popular elastic, ..... implicit-explicit Newmark's method used for the numerical integration of the ...

are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterativemethod, a number of strategies are available for choosing the relaxation parameter...... and the stopping rule. The relaxation parameter can be fixed, or chosen adaptively in each iteration; in the former case we provide a new ‘‘training’’ algorithm that finds the optimal parameter for a given test problem. The stopping rules provided are the discrepancy principle, the monotone error rule, and the NCP...

This paper presents a predictive control algorithm for non-linear systems based on successive linearizations of the non-linear dynamic around a given trajectory. A linear time varying model is obtained and the non-convex constrained optimization problem is transformed into a sequence of locally convex ones. The robustness of the proposed algorithm is addressed adding a convex contractive constraint. To account for linearization errors and to obtain more accurate results an inner iteration loop is added to the algorithm. A simple methodology to obtain an outer bounding-tube for state trajectories is also presented. The convergence of the iterative process and the stability of the closed-loop system are analyzed. The simulation results show the effectiveness of the proposed algorithm in controlling a quadcopter type unmanned aerial vehicle.

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iterationmethod. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

Full Text Available In a paper [Appl. Math. Comput., 188 (2 (2007 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.

Full Text Available Based on the fourth-order method of Liu et al. [10], eighth-order three-step iterativemethods without memory, which are totally free from derivative calculation and reach the optimal efficiency index are presented. The extension of one of the methods for multiple zeros without the knowledge of multiplicity is presented. Further accelerations will be provided through the concept of with memory iterationmethods. Moreover, it is shown by way of illustration that the novel methods are useful on a series of relevant numerical problems when high precision computing is required.

We present an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in vortex particle methods. This is achieved by implementing a penalization of the velocity field using iteration of the penalized vorticity. We show that using the conventional Brinkman penalization method can result in an insufficient enforcement of solid boundaries. The specific problems of the conventional penalization method is discussed and three examples are presented by which the method in its current form has shown to be insufficient to consistently enforce the no-slip boundary condition. These are: the impulsively started flow past a cylinder, the impulsively started flow normal to a flat plate, and the uniformly accelerated flow normal to a flat plate. The iterative penalization algorithm is shown to give significantly improved results compared to the conventional penalization method for each of the presented flow cases.

Comprehensive and complete, this overview provides a single-volume treatment of key algorithms and theories. The author provides clear explanations of all theoretical aspects, with rigorous proof of most results. The two-part treatment begins with the derivation of optimality conditions and discussions of convex programming, duality, generalized convexity, and analysis of selected nonlinear programs. The second part concerns techniques for numerical solutions and unconstrained optimization methods, and it presents commonly used algorithms for constrained nonlinear optimization problems. This g

The dropping damage evaluation for packaging system is essential for safe transportation and storage. A dynamic model of nonlinear cubic-quintic Duffing oscillator for the suspension spring packaging system was proposed. Then, a first-order approximate solution was obtained by applying He’s variable iterationmethod. Based on the results, a damage evaluation equation was derived, which reveals the main controlling physical parameters for damage potential of drop to packaged products concretel...

Full Text Available This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterativemethod is easy to implement and accurate when applied to time-fractional partial differential equations.

Kristiansen (1963) has discussed the convergence of a group of iterativemethods (denoted the Equipoise methods) for the solution of reactor criticality problems. The main result was that even though the methods are said to work satisfactorily in all practical cases, examples of divergence can...

The conventional double sideband (DSB) modulation and direct detection scheme suffers from severer power fading, linear and nonlinear inter-symbol interference (ISI) caused by fiber dispersion and square-law direct detection. The system's frequency response deteriorates at high frequencies owing to the limited device bandwidth. Moreover, the linear and nonlinear ISI is enhanced induced by the bandwidth limited effect. In this paper, an optical tilted filter is used to mitigate the effect of power fading, and improve the high frequency response of bandwidth limited device in Nyquist 4-ary pulse amplitude modulation (4-PAM) system. Furtherly, iterative technique is introduced to mitigate the nonlinear ISI caused by the combined effects of electrical Nyquist filter, limited device bandwidth, optical tilted filter, dispersion, and square-law photo-detection. Thus, the system's frequency response is greatly improved and the delivery distance can be extended.

We construct a modified Bernoulli iterationmethod for solving the quadratic matrix equation AX2+BX+C=0,where A,B and C are square matrices.This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices.Under suitable conditions, we prove the local linear convergence of the Dew method.An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm.In addition,we also describe and analyze the block version of the modified Bernoulli iterationmethod.

of a nonlinear complementarity problem (NCP), which can be solved using an iterative splitting method, such as the projected Gauss–Seidel (PGS) method. We present a novel method for solving the NCP problem by applying a Fletcher–Reeves type nonlinear nonsmooth conjugate gradient (NNCG) type method. We analyze...

to the physical solution, especially for the locations near the blade tip and root where the failure rate of the iterativemethod is high. The stability and accuracy of aerodynamic calculations and optimizations are greatly reduced due to this problem. The intrinsic mechanisms leading to convergence problems......The blade element momentum (BEM) theory is widely used in aerodynamic performance calculations and optimization applications for wind turbines. The fixed point iterativemethod is the most commonly utilized technique to solve the BEM equations. However, this method sometimes does not converge...

The high oating point performance and memory bandwidth of Graphical Processing Units (GPUs) makes them ideal for a large number of computations which often arises in scientic computing, such as matrix operations. GPUs achieve this performance by utilizing massive par- allelism, which requires...... on their applicability for GPUs. We examine published techniques for iterativemethods in interior points methods (IPMs) by applying them to simple test cases, such as a system of masses connected by springs. Iterativemethods allows us deal with the ill-conditioning occurring in the later iterations of the IPM as well...... as to avoid the use of dense matrices, which may be too large for the limited memory capacity of current graphics cards....

In the field of computer vision, camera calibration is a hot issue. For the existing coupled problem of calculating distortion center and the distortion factor in the process of camera calibration, this paper presents an iterative-decreasing calibration method based on regional circle, uses the local area of the circle plate to calculate the distortion center coordinates by iterative declining, and then uses the distortion center to calculate the local area calibration factors. Finally, makes distortion center and the distortion factor for the global optimization. The calibration results show that the proposed method has high calibration accuracy.

Sum of squares (SOS) polynomials have provided a computationally tractable way to deal with inequality constraints appearing in many control problems. It can also act as an approximator in the framework of adaptive dynamic programming. In this paper, an approximate solution to the H∞ optimal control of polynomial nonlinear systems is proposed. Under a given attenuation coefficient, the Hamilton-Jacobi-Isaacs equation is relaxed to an optimization problem with a set of inequalities. After applying the policy iteration technique and constraining inequalities to SOS, the optimization problem is divided into a sequence of feasible semidefinite programming problems. With the converged solution, the attenuation coefficient is further minimized to a lower value. After iterations, approximate solutions to the smallest L₂-gain and the associated H∞ optimal controller are obtained. Four examples are employed to verify the effectiveness of the proposed algorithm.

This paper constructs a proportional-type networked iterative learning control (NILC) scheme for a class of discrete-time nonlinear systems with the stochastic data communication delay within one operation duration and being subject to Bernoulli-type distribution. In the scheme, the communication delayed data is replaced by successfully captured one at the concurrent sampling moment of the latest iteration. The tracking performance of the addressed NILC algorithm is analysed by statistic technique in virtue of mathematical expectation. The analysis shows that, under certain conditions, the expectation of the tracking error measured in the form of 1-norm is asymptotically convergent to zero. Numerical experiments are carried out to illustrate the validity and effectiveness.

This paper presents an efficient method for identification of nonlinear Multi-Input Multi-Output (MIMO) systems in the presence of colored noises. The method studies the multivariable nonlinear Hammerstein and Wiener models, in which, the nonlinear memory-less block is approximated based on arbitrary vector-based basis functions. The linear time-invariant (LTI) block is modeled by an autoregressive moving average with exogenous (ARMAX) model which can effectively describe the moving average noises as well as the autoregressive and the exogenous dynamics. According to the multivariable nature of the system, a pseudo-linear-in-the-parameter model is obtained which includes two different kinds of unknown parameters, a vector and a matrix. Therefore, the standard least squares algorithm cannot be applied directly. To overcome this problem, a Hierarchical Least Squares Iterative (HLSI) algorithm is used to simultaneously estimate the vector and the matrix of unknown parameters as well as the noises. The efficiency of the proposed identification approaches are investigated through three nonlinear MIMO case studies.

Several more general results on the convergence of the Ishikawa iteration procedures with errors for ψ-strong pseudo-contractions and nonlinear operator equations of ψ-strongly accretive type without Lipschitz assumption in Banach spaces are established, as the direct applications, some stability results of the Ishikawa iterationmethods for ψ-strong pseudocontractions and nonlinear operator equations of ψ-strongly accretive type are also given. They unify, improve and extend the recent corresponding results.

Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. This class of matrices provide bases of matrix splittings for iterativemethods. The remarkable feature of iterativemethods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the accelerated overrelaxation method (GAOR) is introduced. Some theories of the AOR method are extended to the generalized method to include a wide class of matrices. The convergence of the new method is derived for Hermitian positive definite matrices. Finally, some examples are given in order to show the superiority of the new method.

Full Text Available The main objective of the current work was to solve the multispecies predator-prey model. The techniques used here were called the variational iterationmethod (VIM and the Adomian decomposition method (ADM. The advantage of this work is twofold. Firstly, the VIM reduces the computational work. Secondly, in comparison with existing techniques, the VIM is an improvement with regard to its accuracy and rapid convergence. The VIM has the advantage of being more concise for analytical and numerical purposes. Comparisons with the exact solution and the fourth-order Runge-Kutta method (RK4 show that the VIM is a powerful method for the solution of nonlinear equations.

The PD-type iterative learning control design of a class of affine nonlinear time-delay systems with external disturbances is considered. Sufficient conditions guaranteeing the convergence of the n-norm of the tracking error are derived. It is shown that the system outputs can be guaranteed to converge to desired trajectories in the absence of external disturbances and output measurement noises. And in the presence of state disturbances and measurement noises, the tracking error will be bounded uniformly. A numerical simulation example is presented to validate the effectiveness of the proposed scheme.

For enhancing heat transfer between primary surface and the environment, utilization of radiating extended surfaces are common. Especially for large temperature differences; variable thermal conductivity has a strong effect on performance of such a surface. In this paper, variational iterationmethod is used to analyze the efficiency of convective straight fins with temperature dependent thermal conductivity. VIM produces analytical expressions for the solution of nonlinear differential equations. In order to show the effectiveness of variational iterationmethod (VIM), the results obtained from VIM analysis is compared with available solutions obtained using Adomian decomposition method (ADM) and the results from finite element analysis. This work assures that VIM is a promising method for the efficiency analysis of convective straight fin problems.

A pseudo-random sequence generator is a basic tool for cryptography. To realize a pseudo-random sequence generator, a new block iteratingmethod using shifter, multiplier,and adder operations has been introduced. By increasing the iteration of the counter and by performing calculations based on the initial value, an approximate pseudo-random sequence was obtained after exchanging bits. The algorithm and the complexity of the generator were introduced. The result obtained from the calculation shows that the self-correlation of the "m" block sequence is two-valued; the block field value is [0,2m - 1 ], and the block period is 2m+8 - 1.

This article proposes to add a simple term to an iterative algebraic equation solver with an order n convergence rate, and to raise the order of convergence to (2n - 1). In particular, a simple algebraic equation solver with the 5th order convergence but uses only 4 function values in each iteration, is described in details. When this scheme is applied to a Newton-Raphson method of the quadratic convergence for a system of algebraic equations, a cubic convergence can be achieved with an low overhead cost of function evaluation that can be ignored as the size of the system increases.

We have developed and numerically evaluated an iterative algorithm for interferogram processing, which includes the Fourier-transform method, the Gerchberg-Papoulis algorithm and Wiener's filter-based regularization used in combination. Using a signal-to-noise ratio of not less than 1, it has been possible to reconstruct the phase of an object field with an accuracy better than 5%.

A parametric nonlinear programming problem in a metric space with an operator equality constraint in a Hilbert space is studied assuming that its lower semicontinuous value function at a chosen individual parameter value has certain subdifferentiability properties in the sense of nonlinear (nonsmooth) analysis. Such subdifferentiability can be understood as the existence of a proximal subgradient or a Fréchet subdifferential. In other words, an individual problem has a corresponding generalized Kuhn-Tucker vector. Under this assumption, a stable sequential Kuhn-Tucker theorem in nondifferential iterative form is proved and discussed in terms of minimizing sequences on the basis of the dual regularization method. This theorem provides necessary and sufficient conditions for the stable construction of a minimizing approximate solution in the sense of Warga in the considered problem, whose initial data can be approximately specified. A substantial difference of the proved theorem from its classical same-named analogue is that the former takes into account the possible instability of the problem in the case of perturbed initial data and, as a consequence, allows for the inherited instability of classical optimality conditions. This theorem can be treated as a regularized generalization of the classical Uzawa algorithm to nonlinear programming problems. Finally, the theorem is applied to the "simplest" nonlinear optimal control problem, namely, to a time-optimal control problem.

Full Text Available In this paper the author proves the algorithms for the existence as well as approximations of the solutions for the initial value problems of nonlinear fractional diﬀerential equations using the operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration principle embodied in the recent hybrid ﬁxed point theorems of Dhage (2014 in a partially ordered normed linear space and the existence and approximations of the solutions of the considered nonlinear fractional diﬀerential equations are obtained under weak mixed partial continuity and partial Lipschitz conditions. Our hypotheses and existence and approximation results are also well illustrated by some numerical examples.

Full Text Available In this paper the author proves the algorithms for the existence as well as approximations of the solutions for the initial value problems of nonlinear fractional diﬀerential equations using the operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration principle embodied in the recent hybrid ﬁxed point theorems of Dhage (2014 in a partially ordered normed linear space and the existence and approximations of the solutions of the considered nonlinear fractional diﬀerential equations are obtained under weak mixed partial continuity and partial Lipschitz conditions. Our hypotheses and existence and approximation results are also well illustrated by some numerical examples.

Children born are susceptible to various diseases such as mumps, chicken pox etc. These diseases are the most common form of infectious diseases. In recent years, scientists have been trying to devise strategies to fight against these diseases. Since vaccination is considered to be the most....... In this article two methods namely Homotopy Perturbation Method (HPM) and Variational IterationMethod (VIM) are employed to compute an approximation to the solution of non-linear system of differential equations governing the problem. The obtained results are compared with those obtained by Adomian Decomposition...

Full Text Available We apply an analytical method called the Optimal Parametric IterationMethod (OPIM to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

Children born are susceptible to various diseases such as mumps, chicken pox etc. These diseases are the most common form of infectious diseases. In recent years, scientists have been trying to devise strategies to fight against these diseases. Since vaccination is considered to be the most....... In this article two methods namely Homotopy Perturbation Method (HPM) and Variational IterationMethod (VIM) are employed to compute an approximation to the solution of non-linear system of differential equations governing the problem. The obtained results are compared with those obtained by Adomian Decomposition...

Full Text Available An extension of the so-called new iterativemethod (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iterationmethod (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

Devices are increasingly vulnerable to soft errors as their feature sizes shrink. Previously, soft error rates were significant primarily in space and high-atmospheric computing. Modern architectures now use features so small at sufficiently low voltages that soft errors are becoming important even at terrestrial altitudes. Due to their large number of components, supercomputers are particularly susceptible to soft errors. Since many large scale parallel scientific applications use iterative linear algebra methods, the soft error vulnerability of these methods constitutes a large fraction of the applications overall vulnerability. Many users consider these methods invulnerable to most soft errors since they converge from an imprecise solution to a precise one. However, we show in this paper that iterativemethods are vulnerable to soft errors, exhibiting both silent data corruptions and poor ability to detect errors. Further, we evaluate a variety of soft error detection and tolerance techniques, including checkpointing, linear matrix encodings, and residual tracking techniques.

This work is to develop a general framework, namely filtered iterative reconstruction (FIR) method, to incorporate analytical reconstruction (AR) method into iterative reconstruction (IR) method, for enhanced CT image quality. Specifically, FIR is formulated as a combination of filtered data fidelity and sparsity regularization, and then solved by proximal forward-backward splitting (PFBS) algorithm. As a result, the image reconstruction decouples data fidelity and image regularization with a two-step iterative scheme, during which an AR-projection step updates the filtered data fidelity term, while a denoising solver updates the sparsity regularization term. During the AR-projection step, the image is projected to the data domain to form the data residual, and then reconstructed by certain AR to a residual image which is in turn weighted together with previous image iterate to form next image iterate. Since the eigenvalues of AR-projection operator are close to the unity, PFBS based FIR has a fast convergenc...

We introduce a general itcrative scheme for angle-limited image reconstruction based on Landwe-ber's method. We derive a representation formula for this scheme and consequently establish its convergence conditions. Our results suggest certain relaxation strategies for an accelerated convergcnce for angle-limited im-age reconstruction in L2-norm comparing with alternative projection methods. The convolution-backprojection algorithm is given for this iterative process.

In this talk the author discusses how object-oriented programming techniques can be used to develop a flexible software package for preconditioned iterativemethods. The ideas described have been used to implement the linear algebra part of Diffpack, which is a collection of C++ class libraries that provides high-level tools for the solution of partial differential equations. In particular, this software package is aimed at rapid development of PDE-based numerical simulators, primarily using finite element methods.

Computed tomography (CT) plays a key role in modern medical system, whether it be for diagnosis or therapy. As an increased risk of cancer development is associated with exposure to radiation, reducing radiation exposure in CT becomes an essential issue. Based on the compressive sensing (CS) theory, iterative based method with total variation (TV) minimization is proven to be a powerful framework for few-view tomographic image reconstruction. Multigrid method is an iterativemethod for solving both linear and nonlinear systems, especially when the system contains a huge number of components. In medical imaging, image background is often defined by zero intensity, thus attaining spatial support of the image, which is helpful for iterative reconstruction. In the proposed method, the image support is not considered as a priori knowledge. Rather, it evolves during the reconstruction process. Based on the CS framework, we proposed a multigrid method with adaptive spatial support constraint. The simultaneous algebraic reconstruction (SART) with TV minimization is implemented for comparison purpose. The numerical result shows: 1. Multigrid method has better performance while less than 60 views of projection data were used, 2. Spatial support highly improves the CS reconstruction, and 3. When few views of projection data were measured, our method performs better than the SART+TV method with spatial support constraint.

The simultaneous inclusion of polynomial complex zeros is a crucial problem in numerical analysis. Rapidly converging algorithms are presented in these notes, including convergence analysis in terms of circular regions, and in complex arithmetic. Parallel circular iterations, where the approximations to the zeros have the form of circular regions containing these zeros, are efficient because they also provide error estimates. There are at present no book publications on this topic and one of the aims of this book is to collect most of the algorithms produced in the last 15 years. To decrease the high computational cost of interval methods, several effective iterative processes for the simultaneous inclusion of polynomial zeros which combine the efficiency of ordinary floating-point arithmetic with the accuracy control that may be obtained by the interval methods, are set down, and their computational efficiency is described. The rate of these methods is of interest in designing a package for the simultaneous ...

Orbital optimization procedure is widely called in electronic structure simulation. To efficiently find the orbital optimization solution, we developed a new second order orbital optimization algorithm, co-iteration augmented Hessian (CIAH) method. In this method, the orbital optimization is embedded in the diagonalization procedure for augmented Hessian (AH) eigenvalue equation. Hessian approximations can be easily employed in this method to improve the computational costs. We numerically performed the CIAH algorithm with SCF convergence of 20 challenging systems and Boys localization of C60 molecule. We found that CIAH algorithm has better SCF convergence and less computational costs than direct inversion iterative subspace (DIIS) algorithm. The numerical tests suggest that CIAH is a stable, reliable and efficient algorithm for orbital optimization problem.

In this paper, the variational iterationmethod (VIM) has been used to investigate the non-linear vibration of single-walled carbon nanotubes (SWCNTs) based on the nonlocal Timoshenko beam theory. The accuracy of results is examined by the fourth-order Runge-Kutta numerical method. Comparison between VIM solutions with numerical results leads to highly accurate solutions. Also, the behavior of deflection and frequency in vibrations of SWCNTs are studied. The results show that frequency of single walled carbon nanotube versus amplitude increases by increasing the values of B.

Devices become increasingly vulnerable to soft errors as their feature sizes shrink. Previously, soft errors primarily caused problems for space and high-atmospheric computing applications. Modern architectures now use features so small at sufficiently low voltages that soft errors are becoming significant even at terrestrial altitudes. The soft error vulnerability of iterative linear algebra methods, which many scientific applications use, is a critical aspect of the overall application vulnerability. These methods are often considered invulnerable to many soft errors because they converge from an imprecise solution to a precise one. However, we show that iterativemethods can be vulnerable to soft errors, with a high rate of silent data corruptions. We quantify this vulnerability, with algorithms generating up to 8.5% erroneous results when subjected to a single bit-flip. Further, we show that detecting soft errors in an iterativemethod depends on its detailed convergence properties and requires more complex mechanisms than simply checking the residual. Finally, we explore inexpensive techniques to tolerate soft errors in these methods.

Iterative algorithms for solving linear systems of equations often involve the solution of a subproblem at each step. This subproblem is usually another linear system of equations. For example, a preconditioned iteration involves the solution of a preconditioner at each step. In this paper, the author considers algorithms for which the subproblem is also solved iteratively. The subproblem is then said to be solved by {open_quotes}inner iterations{close_quotes} while the term {open_quotes}outer iteration{close_quotes} refers to a step of the basic algorithm. The cost of performing an outer iteration is dominated by the solution of the subproblem, and can be measured by the number of inner iterations. A good measure of the total amount of work needed to solve the original problem to some accuracy c is then, the total number of inner iterations. To lower the amount of work, one can consider solving the subproblems {open_quotes}inexactly{close_quotes} i.e. not to full accuracy. Although this diminishes the cost of solving each subproblem, it usually slows down the convergence of the outer iteration. It is therefore interesting to study the effect of solving each subproblem inexactly on the total amount of work. Specifically, the author considers strategies in which the accuracy to which the inner problem is solved, changes from one outer iteration to the other. The author seeks the `optimal strategy`, that is, the one that yields the lowest possible cost. Here, the author develops a methodology to find the optimal strategy, from the set of slowly varying strategies, for some iterative algorithms. This methodology is applied to the Chebychev iteration and it is shown that for Chebychev iteration, a strategy in which the inner-tolerance remains constant is optimal. The author also estimates this optimal constant. Then generalizations to other iterative procedures are discussed.

The large scale linear systems with M-matrices often appear in a wide variety of areas of physical, fluid dynamics and economic sciences. It is reported in [1] that the convergence rate of the IMGS method, with the preconditioner I + Sα, is superior to that of the basic SOR iterativemethod for the M-matrix. This paper considers the preconditioned Jacobi (PJ) method with the preconditioner P = I + Sα + Sβ, and proves theoretically that the convergence rate of the PJ method is better than that of the basic AOR method. Numerical examples are provided to illustrate the main results obtained.

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterativemethods rather than collocation-type methods. Compared to collocation methods, which require mesh refinements to ensure uniform convergence...... with respect to , appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples, including a model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables, and the computation of homoclinic connections in the Fitz...

Full Text Available A model with its conductivity varying highly across a very thin layer will be considered. It is related to a stable phantom model, which is invented to generate a certain apparent conductivity inside a region surrounded by a thin cylinder with holes. The thin cylinder is an insulator and both inside and outside the thin cylinderare filled with the same saline. The injected current can enter only through the holes adopted to the thin cylinder. The model has a high contrast of conductivity discontinuity across the thin cylinder and the thickness of the layer and the size of holes are very small compared to the domain of the model problem. Numerical methods for such a model require a very fine mesh near the thin layer to resolve the conductivity discontinuity. In this work, an efficient numerical method for such a model problem is proposed by employing a uniform mesh, which need not resolve the conductivity discontinuity. The discrete problem is then solved by an iterativemethod, where the solution is improved by solving a simple discrete problem with a uniform conductivity. At each iteration, the right-hand side is updated by integrating the previous iterate over the thin cylinder. This process results in a certain smoothing effect on microscopic structures and our discrete model can provide a more practical tool for simulating the apparent conductivity. The convergence of the iterativemethod is analyzed regarding the contrast in the conductivity and the relative thickness of the layer. In numerical experiments, solutions of our method are compared to reference solutions obtained from COMSOL, where very fine meshes are used to resolve the conductivity discontinuity in the model. Errors of the voltage in L2 norm follow O(h asymptotically and the current density matches quitewell those from the reference solution for a sufficiently small mesh size h. The experimental results present a promising feature of our approach for simulating the apparent

Sensitivity to initial conditions in nonlinear dynamical systems leads to exponential divergence of trajectories that are initially arbitrarily close, and hence to unpredictability. Statistical methods have been found to be helpful in extracting useful information about such systems. In this paper, we review briefly some statistical methods employed in the study of deterministic and stochastic dynamical systems. These include power spectral analysis and aliasing, extreme value statistics and order statistics, recurrence time statistics, the characterization of intermittency in the Sinai disorder problem, random walk analysis of diffusion in the chaotic pendulum, and long-range correlations in stochastic sequences of symbols.

This paper studies non-convex programming problems. It is known that, in statistical inference, many constrained estimation problems may be expressed as convex programming problems. However, in many practical problems, the objective functions are not convex. In this paper, we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems. A two-step iterative algorithm called the alternating iterativemethod is proposed for finding solutions for such problems. The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al. (Biometrika, 72, 465–472 (1983)), Shi N. Z. (J. Multivariate Anal.,50, 282–293 (1994)) and El Barmi H. and Dykstra R. (Ann. Statist., 26, 1878–1893 (1998)).

A nonlinear inversion scheme for the electromagnetic microwave imaging of domains with sparse content is proposed. Scattering equations are constructed using a contrast-source (CS) formulation. The proposed method uses an inexact Newton (IN) scheme to tackle the nonlinearity of these equations. At every IN iteration, a system of equations, which involves the Frechet derivative (FD) matrix of the CS operator, is solved for the IN step. A sparsity constraint is enforced on the solution via thresholded Landweber iterations, and the convergence is significantly increased using a preconditioner that levels the FD matrix\\'s singular values associated with contrast and equivalent currents. To increase the accuracy, the weight of the regularization\\'s penalty term is reduced during the IN iterations consistently with the scheme\\'s quadratic convergence. At the end of each IN iteration, an additional thresholding, which removes small \\'ripples\\' that are produced by the IN step, is applied to maintain the solution\\'s sparsity. Numerical results demonstrate the applicability of the proposed method in recovering sparse and discontinuous dielectric profiles with high contrast values.

To the optimal control problem of affine nonlinear system, based on differential geometry theory, feedback precise linearization was used. Then starting from the simulative relationship between computational structural mechanics and optimal control,multiple-substructure method was inducted to solve the optimal control problem which was linearized. And finally the solution to the original nonlinear system was found. Compared with the classical linearizational method of Taylor expansion, this one diminishes the abuse of error expansion with the enlargement of used region.

are taken to recover a signal/image via the lasso. Solutions of the Q-lasso depend on a tuning parameter γ. In this paper, we obtain basic properties of the solutions as a function of γ. Because of ill posedness, we also apply l1-l2 regularization to the Q-lasso. In addition, we discuss iterativemethods for solving the Q-lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.

In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.

A method is presented to compute the dielectric function for extended systems using linear response time-dependent density functional theory. Localized basis functions with finite support are used to expand both eigenstates and response functions. The electron-energy loss function is directly obtained by an iterative Krylov-subspace method. We apply our method to graphene and silicon and compare it to plane-wave based approaches. Finally, we compute electron-energy loss spectrum of C{sub 60} crystal to demonstrate the merits of the method for molecular crystals, where it will be most competitive.

A method for constructing metric gravity's N-body Lagrangian is developed which uses iterative, liner algebraic euqations which enforce invariance properties of gravity --- exterior effacement, interior effacement, and the time dilation and Lorentz contraction of matter under boosts. The method is demonstrated by obtaining the full 1/c^4 order Lagrangian, and a combination of exterior and interior effacement enforcement permits construction of the full Schwarzschild temporal and spatial metric potentials.

When developing a game it is always valuable to use feedback from players in each iteration, in order to plan the design of the next iteration. However, it can be challenging to devise a simple approach to acquiring information about a player's engagement while playing. In this paper we will thus...... concerning a crowd game which is controlled by smartphones and is intended to be played by audiences in cinemas and at venues with large screens. The case study demonstrates how the approach can be used to help improve the desire to continue when developing a game....... use an evaluation method which focuses on assessing the desire to continue playing as an indicator of player engagement. This feedback can then be applied to detect and prevent any design decisions that would jeopardise a game's level of player engagement. The process is exemplified by a case study...

Full Text Available The dropping damage evaluation for packaging system is essential for safe transportation and storage. A dynamic model of nonlinear cubic-quintic Duffing oscillator for the suspension spring packaging system was proposed. Then, a first-order approximate solution was obtained by applying He’s variable iterationmethod. Based on the results, a damage evaluation equation was derived, which reveals the main controlling physical parameters for damage potential of drop to packaged products concretely. Finally, the dropping damage boundary curves and surfaces for the system were discussed. It was found that decreasing the suspension angle can improve the safe region of the system.

This paper considers dealing with path constraints in the framework of the improved control vector it-eration (CVI) approach. Two available ways for enforcing equality path constraints are presented, which can be di-rectly incorporated into the improved CVI approach. Inequality path constraints are much more difficult to deal with, even for small scale problems, because the time intervals where the inequality path constraints are active are unknown in advance. To overcome the challenge, the l1 penalty function and a novel smoothing technique are in-troduced, leading to a new effective approach. Moreover, on the basis of the relevant theorems, a numerical algo-rithm is proposed for nonlinear dynamic optimization problems with inequality path constraints. Results obtained from the classic batch reactor operation problem are in agreement with the literature reports, and the computational efficiency is also high.

boundary value problems. This note concludes that the method is a modified variational iterationmethod using He’s polynomials. A standard variational iteration algorithm for fractional differential equations is suggested.

Full Text Available Abstract Some convergence conditions on successive over-relaxed (SOR iterativemethod and symmetric SOR (SSOR iterativemethod are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.

Full Text Available We consider a system of nonlinear equations F(x=0. A new iterativemethod for solving this problem numerically is suggested. The analytical discussions of the method are provided to reveal its sixth order of convergence. A discussion on the efficiency index of the contribution with comparison to the other iterativemethods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica.

There are growing applications of quantitative computed tomography for assessment of pulmonary diseases by characterizing lung parenchyma as well as the bronchial tree. Many large multi-center studies incorporating lung imaging as a study component are interested in phenotypes relating airway branching patterns, wall-thickness, and other morphological measures. To our knowledge, there are no fully automated airway tree segmentation methods, free of the need for user review. Even when there are failures in a small fraction of segmentation results, the airway tree masks must be manually reviewed for all results which is laborious considering that several thousands of image data sets are evaluated in large studies. In this paper, we present a CT-based novel airway tree segmentation algorithm using iterative multi-scale leakage detection, freezing, and active seed detection. The method is fully automated requiring no manual inputs or post-segmentation editing. It uses simple intensity based connectivity and a new leakage detection algorithm to iteratively grow an airway tree starting from an initial seed inside the trachea. It begins with a conservative threshold and then, iteratively shifts toward generous values. The method was applied on chest CT scans of ten non-smoking subjects at total lung capacity and ten at functional residual capacity. Airway segmentation results were compared to an expert's manually edited segmentations. Branch level accuracy of the new segmentation method was examined along five standardized segmental airway paths (RB1, RB4, RB10, LB1, LB10) and two generations beyond these branches. The method successfully detected all branches up to two generations beyond these segmental bronchi with no visual leakages.

Nonlinear formulations of the meshless local Petrov-Galerkin (MLPG) method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation are imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.

We present a 3D finite element solver for the nonlinear Poisson-Nernst-Planck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule dynamics inside nanopore sensors. We add to existing numerical approaches by deploying goal-oriented adaptive mesh refinement. To reduce the computation overhead of mesh adaptivity, our error estimator uses the much cheaper Poisson-Boltzmann equation as a simplified model, which is justified on heuristic grounds but shown to work well in practice. To address the nonlinearity in the full PNP-Stokes system, three different linearization schemes are proposed and investigated, with two segregated iterative approaches both outperforming a naive application of Newton's method. Numerical experiments are reported on a real-world nanopore sensor geometry. We also investigate two different models for the interaction of target molecules with the nanopore sensor through the PNP-Stokes equ...

Elastomeric composites have increasingly proved invaluable in commercial technological applications due to their unique mechanical properties, especially their ability to undergo large reversible deformation in response to a variety of stimuli (e.g., mechanical forces, electric and magnetic fields, changes in temperature). Modern advances in organic materials science have revealed that elastomeric composites hold also tremendous potential to enable new high-end technologies, especially as the next generation of sensors and actuators featured by their low cost together with their biocompatibility, and processability into arbitrary shapes. This potential calls for an in-depth investigation of the macroscopic mechanical/physical behavior of elastomeric composites directly in terms of their microscopic behavior with the objective of creating the knowledge base needed to guide their bottom-up design. The purpose of this thesis is to generate a mathematical framework to describe, explain, and predict the macroscopic nonlinear elastic behavior of filled elastomers, arguably the most prominent class of elastomeric composites, directly in terms of the behavior of their constituents --- i.e., the elastomeric matrix and the filler particles --- and their microstructure --- i.e., the content, size, shape, and spatial distribution of the filler particles. This will be accomplished via a combination of novel iterative and variational homogenization techniques capable of accounting for interphasial phenomena and finite deformations. Exact and approximate analytical solutions for the fundamental nonlinear elastic response of dilute suspensions of rigid spherical particles (either firmly bonded or bonded through finite size interphases) in Gaussian rubber are first generated. These results are in turn utilized to construct approximate solutions for the nonlinear elastic response of non-Gaussian elastomers filled with a random distribution of rigid particles (again, either firmly

This paper deals with designing of iterative learning control schemes for uncertain systems with static nonlinearities. More specifically, the nonlinear part is supposed to be sector bounded and system matrices are assumed to range in the polytope of matrices. For systems with such nonlinearities and uncertainties the repetitive process setting is exploited to develop a linear matrix inequality based conditions for computing the feedback and feedforward (learning) controllers. These controllers guarantee acceptable dynamics along the trials and ensure convergence of the trial-to-trial error dynamics, respectively. Numerical examples illustrate the theoretical results and confirm effectiveness of the designed control scheme.

Three-dimensional (3D) subsurface imaging by using inversion of data obtained from the very early time electromagnetic system (VETEM) was discussed. The study was carried out by using the distorted Born iterativemethod to match the internal nonlinear property of the 3D inversion problem. The forward solver was based on the total-current formulation bi-conjugate gradient-fast Fourier transform (BCCG-FFT). It was found that the selection of regularization parameter follow a heuristic rule as used in the Levenberg-Marquardt algorithm so that the iteration is stable.

Crack detection is an important issue in concrete infrastructure. Firstly, the accuracy of crack geometry parameters measurement is directly affected by the extraction accuracy, the same as the accuracy of the detection system. Due to the properties of unpredictability, randomness and irregularity, it is difficult to establish recognition model of crack. Secondly, various image noise, caused by irregular lighting conditions, dark spots, freckles and bump, exerts an influence on the crack detection accuracy. Peak threshold selection method is improved in this paper, and the processing of enhancement, smoothing and denoising is conducted before iterative threshold selection, which can complete the automatic selection of the threshold value in real time and stability.

Multiband signal fusion technique is a practicable and efficient way to improve the range resolution of ISAR image. The classical fusion method estimates the poles of each subband signal by the root-MUSIC method, and some good results were get in several experiments. However, this method is fragile in noise for the proper poles could not easy to get in low signal to noise ratio (SNR). In order to eliminate the influence of noise, this paper propose a matrix pencil algorithm based method to estimate the multiband signal poles. And to deal with mutual incoherent between subband signals, the incoherent parameters (ICP) are predicted through the relation of corresponding poles of each subband. Then, an iterative algorithm which aimed to minimize the 2-norm of signal difference is introduced to reduce signal fusion error. Applications to simulate dada verify that the proposed method get better fusion results at low SNR.

Though the use of the integrated force method for linear investigations is well-recognised, no efforts were made to extend this method to nonlinear structural analysis. This paper presents the attempts to use this method for analysing nonlinear structures. General formulation of nonlinear structural analysis is given. Typically highly nonlinear bench-mark problems are considered. The characteristic matrices of the elements used in these problems are developed and later these structures are analysed. The results of the analysis are compared with the results of the displacement method. It has been demonstrated that the integrated force method is equally viable and efficient as compared to the displacement method.

In recent years, it has turned out that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space V into a sum of subspaces V{sub j}, j = 1{hor_ellipsis}, J resp. of the variational problem on V into auxiliary problems on these subspaces. Here, the author proposes a modified approach to the abstract convergence theory of these additive and multiplicative Schwarz iterativemethods, that makes the relation to traditional iterationmethods more explicit. To this end he introduces the enlarged Hilbert space V = V{sub 0} x {hor_ellipsis} x V{sub j} which is nothing else but the usual construction of the Cartesian product of the Hilbert spaces V{sub j} and use it now in the discretization process. This results in an enlarged, semidefinite linear system to be solved instead of the usual definite system. Then, modern multilevel methods as well as domain decomposition methods simplify to just traditional (block-) iterationmethods. Now, the convergence analysis can be carried out directly for these traditional iterations on the enlarged system, making convergence proofs of multilevel and domain decomposition methods more clear, or, at least, more classical. The terms that enter the convergence proofs are exactly the ones of the classical iterativemethods. It remains to estimate them properly. The convergence proof itself follow basically line by line the old proofs of the respective traditional iterativemethods. Additionally, new multilevel/domain decomposition methods are constructed straightforwardly by now applying just other old and well known traditional iterativemethods to the enlarged system.

Full Text Available This paper presents an adaptive iterative learning control scheme for the output tracking of a class of nonlinear systems with unknown time-varying delays and input saturation nonlinearity. An observer is presented to estimate the states and linear matrix inequality (LMI method is employed for observer design. The assumption of identical initial condition for ILC is relaxed by introducing boundary layer function. The possible singularity problem is avoided by introducing hyperbolic tangent function. The uncertainties with time-varying delays are compensated for by the combination of appropriate Lyapunov-Krasovskii functional and Young’s inequality. Both time-varying and time-invariant radial basis function neural networks are employed to deal with system uncertainties. On the basis of a property of hyperbolic tangent function, the system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapunov-like composite energy function in two cases, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

We consider an Iterated-Subspace Minimization(ISM) method for solving large-scale unconstrained minimization problems. At each major iteration of the method,a two-dimensional manifold, the iterated subspace, is constructed and an approximate minimizer of the objective function in this manifold then determined, and a symmetric rank-one updating is used to solve the inner minimization problem.

Full Text Available Most of the electromagnetic problems can be stated in terms of an inhomogeneous equation Af = g in which A is a differential, integral or integro-differential operator, g in the exitation source and f is the unknown function to be determined. Methods of Moments (MoM is a procedure to solve the equation above and, by means of an appropriate choice of the Basis/Testing (B/T, the problem can be translated into an equivalent linear system even of bigger dimensions. In this work we investigate on how the performances of the major Krylov’s subspace iterative solvers are affected by different choice of these sets of functions. More specifically, as a test case, we consider the algebric linear system of equations obtained by an electrostatic problem of evaluation of the capacitance and electrostatic charge distribution in a cylindrical conductor of finite length. Results are compared in terms of analytical/computational complexity and speed of convergence by exploiting three leading iterativemethods (GMRES, CGS, BibGStab and B/T functions of Pulse/Pulse (P/P and Pulse/Delta (P/D type.

Recent development of Laser-Induced Breakdown Spectroscopy (LIBS) caused that this method is considered as the most promising for future diagnostic applications for characterization of the deposited materials in the International Thermonuclear Experimental Reactor (ITER), which is currently under construction. In this article the basics of LIBS are shortly discussed and the software for spectra analyzing is presented. The main software function is to analyze measured spectra with respect to the certain element lines presence. Some program operation results are presented. Correct results for graphite and aluminum are obtained although identification of tungsten lines is a problem. The reason for this is low tungsten lines intensity, and thus low signal to noise ratio of the measured signal. In the second part artificial neural networks (ANNs) as the next step for LIBS spectra analyzing are proposed. The idea is focused on multilayer perceptron network (MLP) with backpropagation learning method. The potential of ANNs for data processing was proved through application in several LIBS-related domains, e.g. differentiating ancient Greek ceramics (discussed). The idea is to apply an ANN for determination of W, Al, C presence on ITER-like plasma-facing materials.

Two dimensional three temperatures energy equation is a kind of very impor-tant partial differential equation. In general, we discrete such equation with full implicit nine points stencil on Lagrange structured grid and generate a non-linear sparse algebraic equation including nine diagonal lines. This paper will discuss the iterative solver for such non-linear equations. We linearize the equations by fixing the coefficient matrix, and iteratively solve the linearized algebraic equation with Krylov subspace iterativemethod. We have applied the iterativemethod presented in this paper to the code Lared-Ⅰ for numerical simulation of two dimensional threetemperatures radial fluid dynamics, and have obtained efficient results.

A sparsity-regularized Born iterativemethod (BIM) is proposed for efficiently reconstructing two-dimensional piecewise-continuous inhomogeneous dielectric profiles. Such profiles are typically not spatially sparse, which reduces the efficiency of the sparsity-promoting regularization. To overcome this problem, scattered fields are represented in terms of the spatial derivative of the dielectric profile and reconstruction is carried out over samples of the dielectric profile\\'s derivative. Then, like the conventional BIM, the nonlinear problem is iteratively converted into a sequence of linear problems (in derivative samples) and sparsity constraint is enforced on each linear problem using the thresholded Landweber iterations. Numerical results, which demonstrate the efficiency and accuracy of the proposed method in reconstructing piecewise-continuous dielectric profiles, are presented.

Shape-based regularization has proven to be a useful method for delineating objects within noisy images where one has prior knowledge of the shape of the targeted object. When a collection of possible shapes is available, the specification of a shape prior using kernel density estimation is a natural technique. Unfortunately, energy functionals arising from kernel density estimation are of a form that makes them impossible to directly minimize using efficient optimization algorithms such as graph cuts. Our main contribution is to show how one may recast the energy functional into a form that is minimizable iteratively and efficiently using graph cuts.

This paper presents an efficient iterative scheme for the generalized Stokes problem, which arises frequently in the simulation of time-dependent Navier-Stokes equations for incompressible fluid flow. The general form of the linear system is where A = {alpha}M + vT is an n x n symmetric positive definite matrix, in which M is the mass matrix, T is the discrete Laplace operator, {alpha} and {nu} are positive constants proportional to the inverses of the time-step {Delta}t and the Reynolds number Re respectively, and B is the discrete gradient operator of size n x k (k < n). Even though the matrix A is symmetric and positive definite, the system is indefinite due to the incompressibility constraint (B{sup T}u = 0). This causes difficulties both for iterativemethods and commonly used preconditioners. Moreover, depending on the ratio {alpha}/{nu}, A behaves like the mass matrix M at one extreme and the Laplace operator T at the other, thus complicating the issue of preconditioning.

In this paper we address the problem of uncertainty management for robust design, and verification of large dynamic networks whose performance is affected by an equally large number of uncertain parameters. Many such networks (e.g. power, thermal and communication networks) are often composed of weakly interacting subnetworks. We propose intrusive and non-intrusive iterative schemes that exploit such weak interconnections to overcome dimensionality curse associated with traditional uncertainty quantification methods (e.g. generalized Polynomial Chaos, Probabilistic Collocation) and accelerate uncertainty propagation in systems with large number of uncertain parameters. This approach relies on integrating graph theoretic methods and waveform relaxation with generalized Polynomial Chaos, and Probabilistic Collocation, rendering these techniques scalable. We analyze convergence properties of this scheme and illustrate it on several examples.

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, I.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iterate, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix.The igenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient,and the effect of image restoration is r0easonably well.

An adaptive symplectic pseudospectral method based on the dual variational principle is proposed and is successfully applied to solving nonlinear optimal control problems in this paper. The proposed method satisfies the first order necessary conditions of continuous optimal control problems, also the symplectic property of the original continuous Hamiltonian system is preserved. The original optimal control problem is transferred into a set of nonlinear equations which can be solved easily by Newton-Raphson iterations, and the Jacobian matrix is found to be sparse and symmetric. The proposed method, on one hand, exhibits exponent convergence rates when the number of collocation points are increasing with the fixed number of sub-intervals; on the other hand, exhibits linear convergence rates when the number of sub-intervals is increasing with the fixed number of collocation points. Furthermore, combining with the hp method based on the residual error of dynamic constraints, the proposed method can achieve given precisions in a few iterations. Five examples highlight the high precision and high computational efficiency of the proposed method.

The displaced phase center antenna (DPCA) technique is an effective strategy to achieve wide-swath synthetic aperture radar (SAR) imaging with high azimuth resolution.However,traditionally,it requires strict limitation of the pulse repetition frequency (PRF） to avoid non-uniform sampling.Otherwise,any deviation could bring serious ambiguity if the data are directly processed using a matched filter.To break this limitation,a recently proposed spectrum reconstruction method is capable of recovering the true spectrum from the nonuniform samples. However,the performance is sensitive to the selection of the PRF.Sparse regularization based imaging may provide a way to overcome this sensitivity. The existing time-domain method,however,requires a large-scale observation matrix to be built,which brings a high computational cost.In this paper,we propose a frequency domain method,called the iterative spectrum reconstruction method,through integration of the sparse regularization technique with spectrum analysis of the DPCA signal.By approximately expressing the observation in the frequency domain,which is realized via a series of decoupled linear operations,the method performs SAR imaging which is then not directly based on the observation matrix,which reduces the computational cost from O(N2) to O(NlogN) (where N is the number of range cells),and is therefore more efficient than the time domain method. The sparse regularization scheme,realized via a fast thresholding iteration,has been adopted in this method,which brings the robustness of the imaging process to the PRF selection.We provide a series of simulations and ground based experiments to demonstrate the high efficiency and robustness of the method.The simulations show that the new method is almost as fast as the traditional mono-channel algorithm,and works well almost independently of the PRF selection.Consequently,the suggested method can be accepted as a practical and efficient wide-swath SAR imaging technique.

Full-waveform LiDAR (FWL) decomposition results often act as the basis for key LiDAR-derived products, for example canopy height, biomass and carbon pool estimation, leaf area index calculation and under canopy detection. To date, the prevailing method for FWL product creation is the Gaussian Decomposition (GD) based on a non-linear Levenberg-Marquardt (LM) optimization for Gaussian node parameter estimation. GD follows a ;greedy; approach that may leave weak nodes undetected, merge multiple nodes into one or separate a noisy single node into multiple ones. In this manuscript, we propose an alternative decomposition method called Linearly Approximated Iterative Gaussian Decomposition (LAIGD method). The novelty of the LAIGD method is that it follows a multi-step ;slow-and-steady; iterative structure, where new Gaussian nodes are quickly discovered and adjusted using a linear fitting technique before they are forwarded for a non-linear optimization. Two experiments were conducted, one using real full-waveform data from NASA's land, vegetation, and ice sensor (LVIS) and another using synthetic data containing different number of nodes and degrees of overlap to assess performance in variable signal complexity. LVIS data revealed considerable improvements in RMSE (44.8% lower), RSE (56.3% lower) and rRMSE (74.3% lower) values compared to the benchmark GD method. These results were further confirmed with the synthetic data. Furthermore, the proposed multi-step method reduces execution times in half, an important consideration as there are plans for global coverage with the upcoming Global Ecosystem Dynamics Investigation LiDAR sensor on the International Space Station.

We developed dynamic load-balancing algorithms for Particle Simulation Methods (PSM) involving short-range interactions, such as Smoothed Particle Hydrodynamics (SPH), Moving Particle Semi-implicit method (MPS), and Discrete Element method (DEM). These are needed to handle billions of particles modeled in large distributed-memory computer systems. Our method utilizes flexible orthogonal domain decomposition, allowing the sub-domain boundaries in the column to be different for each row. The imbalances in the execution time between parallel logical processes are treated as a nonlinear residual. Load-balancing is achieved by minimizing the residual within the framework of an iterativenonlinear solver, combined with a multigrid technique in the local smoother. Our iterativemethod is suitable for adjusting the sub-domain frequently by monitoring the performance of each computational process because it is computationally cheaper in terms of communication and memory costs than non-iterativemethods. Numerical tests demonstrated the ability of our approach to handle workload imbalances arising from a non-uniform particle distribution, differences in particle types, or heterogeneous computer architecture which was difficult with previously proposed methods. We analyzed the parallel efficiency and scalability of our method using Earth simulator and K-computer supercomputer systems.

Frommer's nonlinear multisplitting methods for solving nonlinear systems of equations are extended to the asynchronous setting. Block methods are extended to include overlap as well. Several specific cases are discussed. Sufficient conditions to guarantee their local convergence are given. A numerical example is presented illustrating the performance of the new approach.

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a simple pendulum, and an approximate expression for its period is obtained. Only one iteration leads to high accuracy of the solutions and the relative error for the approximate period is less than 2% for amplitudes as high as 130{sup 0}. Another important point is that this method provides an analytical expression for the angular displacement as a function of time as the sum of an infinite number of harmonics; although for practical purposes it is sufficient to consider only a finite number of harmonics. We believe that the present study may be a suitable and fruitful exercise for teaching and better understanding perturbation techniques in advanced undergraduate courses on classical mechanics.

Large deflection of a cantilever beam subjected to a tip concentrated load is governed by a non-linear differential equation. Since it is hard to find exact or closed form solutions for this non-linear problem, this paper investigates the aforementioned problem via the differential transformation method (DTM) and the variational iterationmethod (VIM), which are well known approximate analytical solutions. The mathematical formulation is yielded to a non-linear two-point boundary value problem. In this study, we compare the DTM and VIM results, with those of Adomian decomposition method (ADM) and the established numerical solution obtained by the Richardson extrapolation in order to verify the accuracy of the proposed methods. As an important result, it is depicted from tabulated data that the DTM results are more accurate in comparison with those obtained by the VIM and ADM, which is one of the objectives of this article. Moreover, the effects of dimensionless end point load, {alpha} , on the slope of any point along the arc length and the dimensionless vertical and horizontal displacements are illustrated and explained. The results reveal that these methods are very effective and convenient in predicting the solution of such problems, and it is predicted that the DTM and VIM can find a wide application in new engineering problems.

Iterative least square phase-measuring techniques address the phase-shifting interferometry issue of sensitivity to vibrations and scanner nonlinearity. In these techniques the wavefront phase and phase steps are determined simultaneously from a single set of phase-shifted fringe frames where the phase shift does not need to have a nominal value or be a priori precisely known. This method is commonly used in laser interferometers in which the contrast of fringes is constant between frames and across the field. We present step-by-step modifications to the basic iterative least square method. These modifications allow for vibration insensitive measurements in an interferometric system in which fringe contrast varies across a single frame, as well as from frame to frame, due to the limited bandwidth light source and the nonzero numerical aperture of the objective. We demonstrate the efficiency of the new algorithm with experimental data, and we analyze theoretically the degree of contrast variation that this new algorithm can tolerate.

This workshop will focus on methods for solution of compressible Navier-Stokes and Euler equations. In particular, attention will be focused on the interaction between the methods used to solve the non-linear algebraic equations (e.g. full Newton or first order Jacobian) and the resulting large sparse systems. Various types of block and incomplete LU factorization will be discussed, as well as stability issues, and the use of Newton-Krylov methods. These techniques will be demonstrated on a variety of model transonic and supersonic airfoil problems. Applications to industrial CFD problems will also be presented. Experience with the use of C++ for solution of large scale problems will also be discussed. The format for this workshop will be four fifteen minute talks, followed by a roundtable discussion.

According to the refined projection principle advocated by Jia[8], we improve the residual iterationmethod of quadratic eigenvalue problems and propose a refined residual iterationmethod. We study the restarting issue of the method and develop a practical algorithm. Preliminary numerical examples illustrate the efficiency of the method.

Integrated modelling of global environmental change impacts faces the challenge that knowledge from the domains of Natural and Social Science must be integrated. This is complicated by often incompatible terminology and the fact that the interactions between subsystems are usually not fully understood at the start of the project. While a modular modelling approach is necessary to address these challenges, it is not sufficient. The remaining question is how the modelled system shall be cut down into modules. While no generic answer can be given to this question, communication tools can be provided to support the process of modularisation and integration. Along those lines of thought a method for building modular integrated models was developed within the EU project DINAS-COAST and applied to construct a first model, which assesses the vulnerability of the world's coasts to climate change and sea-level-rise. The method focuses on the development of a common language and offers domain experts an intuitive interface to code their knowledge in form of modules. However, instead of rigorously defining interfaces between the subsystems at the project's beginning, an iterative model development process is defined and tools to facilitate communication and collaboration are provided. This flexible approach has the advantage that increased understanding about subsystem interactions, gained during the project's lifetime, can immediately be reflected in the model.

Atmospheric physics is a very complicated natural phenomenon and needs to simplify its basic models for the sea-air oscillator. And it is solved by using the approximate method. The variational iterationmethod is a simple and valid method. In this paper the coupled system for a sea-air oscillator model of interdecadal climate fluctuations is considered. Firstly, through introducing a set of functions, and computing the variations, the Lagrange multipliers are obtained. And then, the generalized expressions of variational iteration are constructed. Finally, through selecting appropriate initial iteration from the iteration expressions, the approximations of solution for the sea-air oscillator model are solved successively.

We consider ill-conditioned symmetric positive definite, Toeplitz systems T{sub n}x = b. If we want to solve such a system iteratively with the conjugate gradient method, we can use band-Toeplitz-preconditioners or Sine-Transform-peconditioners M = S{sub n}{Lambda}S{sub n}, S{sub n} the Sine-Transform-matrix and {Lambda} a diagonal matrix. A Toeplitz matrix T{sub n} = (t{sub i-j)}{sub i}{sup n},{sub j=1} is often related to an underlying function f defined by the coefficients t{sub j}, j = -{infinity},..,-1,0, 1,.., {infinity}. There are four cases, for which we want to determine a preconditioner M: - T{sub n} is related to an underlying function which is given explicitly; - T{sub n} is related to an underlying function that is given by its Fourier coefficients; - T{sub n} is related to an underlying function that is unknown; - T{sub n} is not related to an underlying function. Especially for the first three cases we show how positive definite and effective preconditioners based on the Sine-Transform can be defined for general nonnegative underlying function f. To define M, we evaluate or estimate the values of f at certain positions, and build a Sine-transform matrix with these values as eigenvalues. Then, the spectrum of the preconditioned system is bounded from above and away from zero.

Full Text Available This paper presents a new framework for the identification of mechanics-based nonlinear finite element (FE models of civil structures using Bayesian methods. In this approach, recursive Bayesian estimation methods are utilized to update an advanced nonlinear FE model of the structure using the input-output dynamic data recorded during an earthquake event. Capable of capturing the complex damage mechanisms and failure modes of the structural system, the updated nonlinear FE model can be used to evaluate the state of health of the structure after a damage-inducing event. To update the unknown time-invariant parameters of the FE model, three alternative stochastic filtering methods are used: the extended Kalman filter (EKF, the unscented Kalman filter (UKF, and the iterated extended Kalman filter (IEKF. For those estimation methods that require the computation of structural FE response sensitivities with respect to the unknown modeling parameters (EKF and IEKF, the accurate and computationally efficient direct differentiation method (DDM is used. A three-dimensional five-story two-by-one bay reinforced concrete (RC frame is used to illustrate the performance of the framework and compare the performance of the different filters in terms of convergence, accuracy, and robustness. Excellent estimation results are obtained with the UKF, EKF, and IEKF. Because of the analytical linearization used in the EKF and IEKF, abrupt and large jumps in the estimates of the modeling parameters are observed when using these filters. The UKF slightly outperforms the EKF and IEKF.

A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0-th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1-st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example,namely, the two-dimensional Navier-Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.

Full Text Available We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.

In this paper, explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with time-space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV-Burgers equation.

Full Text Available Soil water transport in small, humid, upland catchments is often dominated by subsurface stormflow. Recent studies of this process suggest that at the plot scale, generation of transient saturation may be governed by threshold behavior, and that transient saturation is a prerequisite for lateral flow. The interaction between these plot scale processes yields complex behavior at the hillslope scale. We argue that this complexity should be incorporated into our models. We take an iterative approach to developing our model, starting with a very simple representation of hillslope rainfall-runoff. Next, we design new virtual experiments with which we test our model, while adding more structural complexity. In this study, we present results from three such development cycles, corresponding to three different hillslope-scale, lumped models. Model1 is a linear tank model, which assumes transient saturation to be homogeneously distributed over the hillslope. Model2 assumes transient saturation to be heterogeneously distributed over the hillslope, and that the spatial distribution of the saturated zone does not vary with time. Model3 assumes that transient saturation is heterogeneous both in space and in time. We found that the homogeneity assumption underlying Model1 resulted in hillslope discharge being too steep during the first part of the rising limb, but not steep enough on the second part. Also, peak height was underestimated. The additional complexity in Model2 improved the simulations in terms of the fit, but not in terms of the dynamics. The threshold-based Model3 captured most of the hydrograph dynamics (Nash-Sutcliffe efficiency of 0.98. After having assessed our models in a lumped setup, we then compared Model1 to Model3 in a spatially explicit setup, and evaluated what patterns of subsurface flow were possible with model elements of each type. We found

Full Text Available Continuing from the works of Li et al. (2014, Li (2007, and Kincaid et al. (2000, we present more generalizations and modifications of iterativemethods for solving large sparse symmetric and nonsymmetric indefinite systems of linear equations. We discuss a variety of iterativemethods such as GMRES, MGMRES, MINRES, LQ-MINRES, QR MINRES, MMINRES, MGRES, and others.

An extremely simple and elementary but rigorous derivation of exact solutions of partial differential equations in different dimensions with variable coefficients is given using the variational iterationmethod. The efficiency of the considered method is illustrated by some examples. The results show that the proposed iteration technique, without linearization or small perturbation, is very effective and convenient.

Full Text Available We introduce a new general iterativemethod by using the -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterativemethod is established under some certain control conditions. Our results improve and extend the results announced by many others.

A quadratically convergent Newton-type iterative scheme is proposed for approximating the generalized Drazin inverse bd of the Banach algebra element b. Further, its extension into the form of the hyperpower iterativemethod of arbitrary order p$\\leq$2 is presented. Convergence criteria along with the estimation of error bounds in the computation of bd are discussed. Convergence results confirms the high order convergence rate of the proposed iterative scheme.

Full Text Available We introduce an iterativemethod for finding a common element of the set of solutions of equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed points of finite many nonexpansive mappings. We prove strong convergence of the iterative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for the minimization problem.

For a general formulation of hybrid inverse problems in impedance tomography the Picard and Newton iterative schemes are adapted and four iterative reconstruction algorithms are developed. The general problem formulation includes several existing hybrid imaging modalities such as current density...... impedance imaging, magnetic resonance electrical impedance tomography, and ultrasound modulated electrical impedance tomography, and the unified approach to the reconstruction problem encompasses several algorithms suggested in the literature. The four proposed algorithms are implemented numerically in two...... be based on a theoretical analysis of the underlying inverse problem....

A general form of a distributed feedback control algorithm based on the speed-gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions. This article is part of the themed issue 'Horizons of cybernetical physics'.

Model reduction technique is usually employed in model updating process,In this paper,a new model updating method named as cross-model cross-frequency response function (CMCF) method is proposed and a new iterativemethod associating the model updating method with the model reduction technique is investigated.The new model updating method utilizes the frequency response function to avoid the modal analysis process and it does not need to pair or scale the measured and the analytical frequency response function,which could greatly increase the number of the equations and the updating parameters.Based on the traditional iterativemethod,a correction term related to the errors resulting from the replacement of the reduction matrix of the experimental model with that of the finite element model is added in the new iterativemethod.Comparisons between the traditional iterativemethod and the proposed iterativemethod are shown by model updating examples of solar panels,and both of these two iterativemethods combine the CMCF method and the succession-level approximate reduction technique.Results show the effectiveness of the CMCF method and the proposed iterativemethod.

Full Text Available Ultrasound tomography offers the potential for detecting of very small tumors whose sizes are smaller than the wavelength of the incident pressure wave without ionizing radiation. Based on inverse scattering technique, this imaging modality uses some material properties such as sound contrast and attenuation in order to detect small objects. One of the most commonly used methods in ultrasound tomography is the Distorted Born IterativeMethod (DBIM. The compressed sensing technique was applied in the DBIM as a promising approach for the image reconstruction quality improvement. Nevertheless, the random measurement configuration of transducers in this method is very difficult to set up in practice. Therefore, in this paper, we take advantages of simpler sparse uniform measurement configuration set-up of transducers and high-quality image reconstruction of 1 non-linear regularization in sparse scattering domain. The simulation results demonstrate the high performance of the proposed approach in terms of tremendously reduced total runtime and normalized error.

A new perturbation-like technique called linearization and correction method is proposed. Contrary to the traditional perturbation techniques, the present theory does not assume that the solution is expressed in the form of a power series of small parameter. To obtain an asymptotic solution of nonlinear system, the technique first searched for a solution for the linearized system, then a correction was added to the linearized solution. So the obtained results are uniformly valid for both weakly and strongly nonlinear equations.

Highlights: ► The product of exponential (POE) formula for error modeling of hybrid robot. ► Differential Evolution (DE) algorithm for parameter identification. ► Simulation results are given to verify the effectiveness of the method. -- Abstract: This paper focuses on the kinematic calibration of a 10 degree-of-freedom (DOF) redundant serial–parallel hybrid robot to improve its accuracy. The robot was designed to perform the assembling and repairing tasks of the vacuum vessel (VV) of the international thermonuclear experimental reactor (ITER). By employing the product of exponentials (POEs) formula, we extended the POE-based calibration method from serial robot to redundant serial–parallel hybrid robot. The proposed method combines the forward and inverse kinematics together to formulate a hybrid calibration method for serial–parallel hybrid robot. Because of the high nonlinear characteristics of the error model and too many error parameters need to be identified, the traditional iterative linear least-square algorithms cannot be used to identify the parameter errors. This paper employs a global optimization algorithm, Differential Evolution (DE), to identify parameter errors by solving the inverse kinematics of the hybrid robot. Furthermore, after the parameter errors were identified, the DE algorithm was adopted to numerically solve the forward kinematics of the hybrid robot to demonstrate the accuracy improvement of the end-effector. Numerical simulations were carried out by generating random parameter errors at the allowed tolerance limit and generating a number of configuration poses in the robot workspace. Simulation of the real experimental conditions shows that the accuracy of the end-effector can be improved to the same precision level of the given external measurement device.

Full Text Available We introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

A method for conducting nonlinear electrochemical impedance spectroscopy. The method includes quantifying the nonlinear response of an electrochemical system by measuring higher-order current or voltage harmonics generated by moderate-amplitude sinusoidal current or voltage perturbations. The method involves acquisition of the response signal followed by time apodization and fast Fourier transformation of the data into the frequency domain, where the magnitude and phase of each harmonic signal can be readily quantified. The method can be implemented on a computer as a software program.

A method for conducting nonlinear electrochemical impedance spectroscopy. The method includes quantifying the nonlinear response of an electrochemical system by measuring higher-order current or voltage harmonics generated by moderate-amplitude sinusoidal current or voltage perturbations. The method involves acquisition of the response signal followed by time apodization and fast Fourier transformation of the data into the frequency domain, where the magnitude and phase of each harmonic signal can be readily quantified. The method can be implemented on a computer as a software program.

In this article, two powerful analytical methods called the variational iterationmethod (VIM) and the variational homotopy perturbation method (VHPM) are introduced to obtain the exact and the numerical solutions of the (2+1)-dimensional Korteweg-de Vries-Burgers (KdVB) equation and the (1+1)-dimensional Sharma-Tasso-Olver equation. The main objective of the present article is to propose alternative methods of solutions, which avoid linearization and physical unrealistic assumptions. The results show that these methods are very efficient, convenient and can be applied to a large class of nonlinear problems. (orig.)

To solve nonlinear complementarity problems (NCP), at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the LogarithmicQuadratic Proximal (LQP) method solves a system of nonlinear equations (LQP system). This paper presents a practical LQP method-based prediction-correction method for NCP.The predictor is obtained via solving the LQP system approximately under significantly relaxed restriction, and the new iterate (the corrector) is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.

Full Text Available We introduce the general iterativemethods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

In particle transport applications there are numerous physical constructs in which heterogeneities are randomly distributed. The quantity of interest in these problems is the ensemble average of the flux, or the average of the flux over all possible material 'realizations.' The Levermore-Pomraning closure assumes Markovian mixing statistics and allows a closed, coupled system of equations to be written for the ensemble averages of the flux in each material. Generally, binary statistical mixtures are considered in which there are two (homogeneous) materials and corresponding coupled equations. The solution process is iterative, but convergence may be slow as either or both materials approach the diffusion and/or atomic mix limits. A three-part acceleration scheme is devised to expedite convergence, particularly in the atomic mix-diffusion limit where computation is extremely slow. The iteration is first divided into a series of 'inner' material and source iterations to attenuate the diffusion and atomic mix error modes separately. Secondly, atomic mix synthetic acceleration is applied to the inner material iteration and S{sup 2} synthetic acceleration to the inner source iterations to offset the cost of doing several inner iterations per outer iteration. Finally, a Krylov iterative solver is wrapped around each iteration, inner and outer, to further expedite convergence. A spectral analysis is conducted and iteration counts and computing cost for the new two-step scheme are compared against those for a simple one-step iteration, to which a Krylov iterativemethod can also be applied.

Full Text Available This paper proposes a new, simple and efficient method for nonlinear simulation of arch dam cracking from the construction period to the operation period, which takes into account the arch dam construction process and temperature loads. In the calculation mesh, the contact surface of pair nodes is located at places on the arch dam where cracking is possible. A new effective iterativemethod, the mixed finite element method for friction-contact problems, is improved and used for nonlinear simulation of the cracking process. The forces acting on the structure are divided into two parts: external forces and contact forces. The displacement of the structure is chosen as the basic variable and the nodal contact force in the possible contact region of the local coordinate system is chosen as the iterative variable, so that the nonlineariterative process is only limited within the possible contact surface and is much more economical. This method was used to simulate the cracking process of the Shuanghe Arch Dam in Southwest China. In order to prove the validity and accuracy of this method and to study the effect of thermal stress on arch dam cracking, three schemes were designed for calculation. Numerical results agree with actual measured data, proving that it is feasible to use this method to simulate the entire process of nonlinear arch dam cracking.

Full Text Available The local fractional variational iterationmethod for local fractional Laplace equation is investigated in this paper. The operators are described in the sense of local fractional operators. The obtained results reveal that the method is very effective.

An iterative separation approach, i.e. source signals are extracted and removed one by one, is proposed for multichannel blind deconvolution of colored signals. Each source signal is extracted in two stages: a filtered version of the source signal is first obtained by solving the generalized eigenvalue problem, which is then followed by a single channel blind deconvolution based on ensemble learning. Simulation demonstrates the capability of the approach to perform efficient mutichannel blind deconvolution.

Full Text Available The complex dynamical analysis of the parametric fourth-order Kim’s iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones.

The complex dynamical analysis of the parametric fourth-order Kim's iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).

This project supported the Copper Mountain Conference on Multigrid and IterativeMethods, held from 2007 to 2015, at Copper Mountain, Colorado. The subject of the Copper Mountain Conference Series alternated between Multigrid Methods in odd-numbered years and IterativeMethods in even-numbered years. Begun in 1983, the Series represents an important forum for the exchange of ideas in these two closely related fields. This report describes the Copper Mountain Conference on Multigrid and IterativeMethods, 2007-2015. Information on the conference series is available at http://grandmaster.colorado.edu/~copper/.

The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.

Full Text Available The optimal pointwise control of the KdV equation is investigated with an objective of minimizing a given performance measure. The performance measure is specified as a quadratic functional of the final state and velocity functions along with the energy due to open- and closed-loop controls. The minimization of the performance measure over the controls is subjected to the KdV equation with periodic boundary conditions and appropriate initial condition. In contrast to standard optimal control or variational methods, a direct control parameterization is used in this study which presents a distinct approach toward the solution of optimal control problems. The method is based on finite terms of Fourier series approximation of each time control variable with unknown Fourier coefficients and frequencies. He's variational iterationmethod for the nonlinear partial differential equations is applied to the problem and thus converting the optimal control of lumped parameter systems into a mathematical programming. A numerical simulation is provided to exemplify the proposed method.

An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iterationmethod (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.

A numerical method for simulating nonlinear fluid-rigid structure interaction problems is developed. The structure is assumed to undergo large rigid body motions and the fluid flow is governed by nonlinear, viscous or non-viscous, field equations with nonlinear boundary conditions applied to the free surface and fluid-solid interaction interfaces. An Arbitrary-Lagrangian-Eulerian (ALE) mesh system is used to construct the numerical model. A multi-block numerical scheme of study is adopted allowing for the relative motion between moving overset grids, which are independent of one another. This provides a convenient method to overcome the difficulties in matching fluid meshes with large solid motions. Nonlinear numerical equations describing nonlinear fluid-solid interaction dynamics are derived through a numerical discretization scheme of study. A coupling iteration process is used to solve these numerical equations. Numerical examples are presented to demonstrate applications of the model developed.

Full Text Available In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iterationmethod and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

Full Text Available It is well known that SOR iterativemethods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices. However, the same can not be true in case of those iterativemethods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices. This paper proposes some necessary and sufficient conditions such that SOR iterativemethods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

Eigenproblems resulting from the use of localised basis functions (typically Gaussian or Slater type orbitals) in density functional electronic-structure calculations are often solved using direct linear algebra. A full implementation is presented built around an iterativemethod known as 'residual minimisation—direct inversion of the iterative subspace' (RM-DIIS) to be used to solve many similar eigenproblems in a self-consistency cycle. The method is more efficient than direct methods and exhibits superior scaling on parallel supercomputers.

The preconditioned Gauss-Seidel type iterativemethod for solving linear systems, with the proper choice of the preconditioner, is presented. Convergence of the preconditioned method applied to Z-matrices is discussed. Also the optimal parameter is presented. Numerical results show that the proper choice of the preconditioner can lead to effective by the preconditioned Gauss-Seidel type iterativemethods for solving linear systems.

Full Text Available The solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iterationmethod. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equations. Application of variational iteration technique to this problem has shown the rapid convergence of the sequence constructed by this method to the exact solution.

In this paper, Moose scheme is used for frequency offset estimation in OFDMA uplink systems due to that the signals from different users can be easily distinguished in frequency domain. However, differential multiple access interference (MAI) will deteriorate the frequency offset estimation performances,especially in interleaved OFDMA system. Analysis and simulation results manifest that frequency offset estimation by Moose scheme in block OFDMA system is more robust than that in interleaved OFDMA system. And an iterative interference cancellation method has been proposed to suppress the differential MAI interference for interleaved OFDMA system, in which Moose scheme is the special case of the number of iteration is equal to one. Simulation results demonstrate that the proposed method can improve the performance with the increase of the number of iterations. In consideration of the performance and complexity,the proposed method with two iterations is selected. And the full comparison results of the proposed iterativemethod with two iterations and that with one iteration (conventional Moose scheme) are given in the paper, which sufficiently demonstrate that the performance gain can be obtained by the interference cancellation operation in interleaved OFDMA system.

A new algorithm called homotopy iterationmethod based on the homotopy function is studied and improved. By the improved homotopy iterationmethod, polynomial systems with high order and deficient can be solved fast and efficiently comparing to the original homotopy iterationmethod. Numerical examples for the ninepoint path synthesis of four-bar linkages show the advantages and efficiency of the improved homotopy iterationmethod.

Laplace transform techniques have been widely used to solve linear, transient field problems. A transform-based algorithm enables calculation of the response at selected times of interest without the need for stepping in time as required by conventional time integration schemes. The elimination of time stepping can substantially reduce computer time when transform techniques are implemented in a numerical finite element program. The coupling of transform techniques with spatial discretization techniques such as the finite element method has resulted in what are known as transfinite element methods. Recently attempts have been made to extend the transfinite element method to solve nonlinear, transient field problems. This paper examines the theoretical basis and numerical implementation of one such algorithm, applied to nonlinear heat transfer problems. The problem is linearized and solved by requiring a numerical iteration at selected times of interest. While shown to be acceptable for weakly nonlinear problems, this algorithm is ineffective as a general nonlinear solution method.

Full Text Available We present recent results concerning the application of the monotone method for studying existence and uniqueness of solutions to general first-order nonlinear nonlocal hyperbolic problems. The limitations of comparison principles for such nonlocal problems are discussed. To overcome these limitations, we introduce new definitions for upper and lower solutions.

The first part of this Review describes a few of the main methods that have been employed in non-linear time series analysis with special reference to biological applications (biomechanics). The second part treats the physical basis of posturogram data (human balance) and EMG (electromyography, a measure of muscle activity).

A compound neural network is utilized to identify the dynamic nonlinear system.This network is composed of two parts: one is a linear neural network,and the other is a recurrent neural network.Based on the inverse theory a compound inverse control method is proposed.The controller has also two parts:a linear controller and a nonlinear neural network controller.The stability condition of the closed-loop neural network-based compound inverse control system is demonstrated based on the Lyapunov theory.Simulation studies have shown that this scheme is simple and has good control accuracy and robustness.

Using the asymptotic iterationmethod (AIM) we obtain the spectrum of the Klein-Gordon equation for some choices of scalar and vector potentials. In particular, it is shown that the AIM exactly reproduces the spectrum of some solvable potentials.

Full Text Available He's variational iterationmethod (VIM is used for solving space and time fractional telegraph equations. Numerical examples are presented in this paper. The obtained results show that VIM is effective and convenient.

Experimental data reveals that attenuation is an important phenomenon in medical ultrasound. Attenuation is particularly important for medical applications based on nonlinear acoustics, since higher harmonics experience higher attenuation than the fundamental. Here, a method is presented to accurately solve the wave equation for nonlinear acoustic media with spatially inhomogeneous attenuation. Losses are modeled by a spatially dependent compliance relaxation function, which is included in the Westervelt equation. Introduction of absorption in the form of a causal relaxation function automatically results in the appearance of dispersion. The appearance of inhomogeneities implies the presence of a spatially inhomogeneous contrast source in the presented full-wave method leading to inclusion of forward and backward scattering. The contrast source problem is solved iteratively using a Neumann scheme, similar to the iterativenonlinear contrast source (INCS) method. The presented method is directionally independent and capable of dealing with weakly to moderately nonlinear, large scale, three-dimensional wave fields occurring in diagnostic ultrasound. Convergence of the method has been investigated and results for homogeneous, lossy, linear media show full agreement with the exact results. Moreover, the performance of the method is demonstrated through simulations involving steered and unsteered beams in nonlinear media with spatially homogeneous and inhomogeneous attenuation.

在凸度量空间内，引入了非线性拟压缩映射序列和改进的Ishikawa型迭代序列，证明了改进的Ishikawa型迭代序列收敛于非线性拟压缩映射序列的唯一公共不动点。%The notion of nonlinear qusi-contraction mappings squence and improved Ishikawa-type iteration are introduced in convex matric space.The result that the improved lshikawa-type iteration sequence converges to unique common fixed point of nonlinear quasi-contraction-mappings sequence is also given.

Full Text Available The variational iterationmethod (VIM is applied to solve singular perturbation initial value problems with delays (SPIVPDs. Some convergence results of VIM for solving SPIVPDs are given. The obtained sequence of iterates is based on the use of general Lagrange multipliers; the multipliers in the functionals can be identified by the variational theory. Moreover, the numerical examples show the efficiency of the method.

In this paper,we consider the existence of symmetric solutions to a nonlinear second order multi-point boundary value problem,and establish corresponding iterative schemes based on the monotone iterativemethod.

We present a comparison of a number of iterative solvers of linear systems of equations for obtaining the fermion propagator in lattice QCD. In particular, we consider chirally invariant overlap and chirally improved Wilson (maximally) twisted mass fermions. The comparison of both formulations of lattice QCD is performed at four fixed values of the pion mass between 230 MeV and 720 MeV. For overlap fermions we address adaptive precision and low mode preconditioning while for twisted mass fermions we discuss even/odd preconditioning. Taking the best available algorithms in each case we find that calculations with the overlap operator are by a factor of 30-120 more expensive than with the twisted mass operator. (orig.)

We present a comparison of a number of iterative solvers of linear systems of equations for obtaining the fermion propagator in lattice QCD. In particular, we consider chirally invariant overlap and chirally improved Wilson (maximally) twisted mass fermions. The comparison of both formulations of lattice QCD is performed at four fixed values of the pion mass between 230 MeV and 720 MeV. For overlap fermions we address adaptive precision and low mode preconditioning while for twisted mass fermions we discuss even/odd preconditioning. Taking the best available algorithms in each case we find that calculations with the overlap operator are by a factor of 30-120 more expensive than with the twisted mass operator. (orig.)

The goal of the work presented is the development of effective iterative techniques for large-scale inverse or parameter estimation problems. In this extended abstract, a detailed description of the mathematical framework in which the authors view these problem is presented, followed by an outline of the ideas and algorithms developed. Distributed parameter estimation problems often arise in mathematical modeling with partial differential equations. They can be viewed as inverse problems; the `forward problem` is that of using the fully specified model to predict the behavior of the system. The inverse or parameter estimation problem is: given the form of the model and some observed data from the system being modeled, determine the unknown parameters of the model. These problems are of great practical and mathematical interest, and the development of efficient computational algorithms is an active area of study.

The harmonic balance (HB) method as an analytical approach is widely used for nonlinear oscillators, in which the initial conditions are generally simplified by setting velocity or displacement to be zero. Based on HB, we establish a new theory to address nonlinear conservative systems with arbitrary initial conditions, and deduce a set of over-determined algebraic equations. Since these deduced algebraic equations are not solved directly, a minimization problem is constructed instead and an iterative algorithm is employed to seek the minimization point. Taking Duffing and Duffing-harmonic equations as numerical examples, we find that these attained solutions are not only with high degree of accuracy, but also uniformly valid in the whole solution domain.

Tri-level optimization problems are optimization problems with three nested hierarchical structures, where in most cases conflicting objectives are set at each level of hierarchy. Such problems are common in management, engineering designs and in decision making situations in general, and are known to be strongly NP-hard. Existing solution methods lack universality in solving these types of problems. In this paper, we investigate a tri-level programming problem with quadratic fractional objective functions at each of the three levels. A solution algorithm has been proposed by applying fuzzy goal programming approach and by reformulating the fractional constraints to equivalent but non-fractional non-linear constraints. Based on the transformed formulation, an iterative procedure is developed that can yield a satisfactory solution to the tri-level problem. The numerical results on various illustrative examples demonstrated that the proposed algorithm is very much promising and it can also be used to solve larger-sized as well as n-level problems of similar structure.

Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlineariterativemethods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlineariteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

To solve vehicle routing problem with different fleets, two methodologies are developed. The first methodology adopts twophase strategy. In the first phase, the improved savings method is used to assign customers to appropriate vehicles. In the second phase, the iterated dynasearch algorithm is adopted to route each selected vehicle with the assigned customers. The iterated dynasearch algorithm combines dynasearch algorithm with iterated local search algorithm based on random kicks. The second methodplogy adopts the idea of cyclic transfer which is performed by using dynamic programming algorithm, and the iterated dynasearch algorithm is also embedded in it. The test results show that both methodologies generate better solutions than the traditional method, and the second methodology is superior to the first one.

A Newton iteration-based interval uncertainty analysis method (NI-IUAM) is proposed to analyze the propagating effect of interval uncertainty in multidisciplinary systems. NI-IUAM decomposes one multidisciplinary system into single disciplines and utilizes a Newton iteration equation to obtain the upper and lower bounds of coupled state variables at each iterative step. NI-IUAM only needs to determine the bounds of uncertain parameters and does not require specific distribution formats. In this way, NI-IUAM may greatly reduce the necessity for raw data. In addition, NI-IUAM can accelerate the convergence process as a result of the super-linear convergence of Newton iteration. The applicability of the proposed method is discussed, in particular that solutions obtained in each discipline must be compatible in multidisciplinary systems. The validity and efficiency of NI-IUAM is demonstrated by both numerical and engineering examples.

Reputation is a valuable asset in online social lives and it has drawn increased attention. How to evaluate user reputation in online rating systems is especially significant due to the existence of spamming attacks. To address this issue, so far, a variety of methods have been proposed, including network-based methods, quality-based methods and group-based ranking method. In this paper, we propose an iterative group-based ranking (IGR) method by introducing an iterative reputation-allocation process into the original group-based ranking (GR) method. More specifically, users with higher reputation have higher weights in dominating the corresponding group sizes. The reputation of users and the corresponding group sizes are iteratively updated until they become stable. Results on two real data sets suggest that the proposed IGR method has better performance and its robustness is considerably improved comparing with the original GR method. Our work highlights the positive role of users' grouping behavior towards...

Implicit Runge-Kutta method is highly accurate and stable for stiff initial value prob-lem. But the iteration technique used to solve implicit Runge-Kutta method requires lots of computational efforts. In this paper, we extend the Parallel Diagonal Iterated Runge-Kutta(PDIRK) methods to delay differential equations(DDEs). We give the convergence region of PDIRK methods, and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector method. Finally, we analysis the speed-up factor through a numerical experiment. The results show that the PDIRK methods to DDEs are efficient.

An important research topic for prospecting seismology is to provide a fast accurate velocity model from pre-stack depth migration. Aiming at such a problem, we propose a quadratic precision generalized nonlinear global optimization migration velocity inversion. First we discard the assumption that there is a linear relationship between residual depth and residual velocity and propose a velocity model correction equation with quadratic precision which enables the velocity model from each iteration to approach the real model as quickly as possible. Second, we use a generalized nonlinear inversion to get the global optimal velocity perturbation model to all traces. This method can expedite the convergence speed and also can decrease the probability of falling into a local minimum during inversion. The synthetic data and Marmousi data examples show that our method has a higher precision and needs only a few iterations and consequently enhances the practicability and accuracy of migration velocity analysis (MVA) in complex areas.

In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new meth

A new formulation for calculating the steady-state responses of multiple-degree-of-freedom (MDOF) non-linear dynamic systems due to harmonic excitation is developed. This is aimed at solving multi-dimensional nonlinear systems using linear equations. Nonlinearity is parameterised by a set of 'non-linear control parameters' such that the dynamic system is effectively linear for zero values of these parameters and nonlinearity increases with increasing values of these parameters. Two sets of linear equations which are formed from a first-order truncated Taylor series expansion are developed. The first set of linear equations provides the summation of sensitivities of linear system responses with respect to non-linear control parameters and the second set are recursive equations that use the previous responses to update the sensitivities. The obtained sensitivities of steady-state responses are then used to calculate the steady state responses of non-linear dynamic systems in an iterative process. The application and verification of the method are illustrated using a non-linear Micro-Electro-Mechanical System (MEMS) subject to a base harmonic excitation. The non-linear control parameters in these examples are the DC voltages that are applied to the electrodes of the MEMS devices.

To improve the performance of Newton`s method for the solution of systems of nonlinear equations a modification to the Newton iteration is implemented. The modified step is taken as a linear combination of Newton step and steepest descent directions. In the paper we describe how the coefficients of the combination can be generated to make effective use of the two component steps. Numerical results that show the usefulness of the combined modification are presented.

K. Nakajo and W. Takahashi in 2003 proved the strong convergence theorems for nonexpansive mappings, nonexpansive semigroups, and proximal point algorithm for zero point of monotone operators in Hilbert spaces by using the hybrid method in mathematical programming. The purpose of this paper is to modify the hybrid iterationmethod of K. Nakajo and W. Takahashi through the monotone hybrid method, and to prove strong convergence theorems. The convergence rate of iteration process of the monotone hybrid method is faster than that of the iteration process of the hybrid method of K. Nakajo and W. Takahashi. In the proofs in this article, Cauchy sequence method is used to avoid the use of the demiclosedness principle and Opial's condition.

In this paper, we aim to solve the infinite-time optimal tracking control problem for a class of discrete-time nonlinear systems using the greedy heuristic dynamic programming (HDP) iteration algorithm. A new type of performance index is defined because the existing performance indexes are very difficult in solving this kind of tracking problem, if not impossible. Via system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then, the greedy HDP iteration algorithm is introduced to deal with the regulation problem with rigorous convergence analysis. Three neural networks are used to approximate the performance index, compute the optimal control policy, and model the nonlinear system for facilitating the implementation of the greedy HDP iteration algorithm. An example is given to demonstrate the validity of the proposed optimal tracking control scheme.

This paper will focus on recent work in two widely used industrial applications codes with iterativemethods. The ANSYS program, a general purpose finite element code widely used in structural analysis applications, has now added an iterative solver option. Some results are given from real applications comparing performance with the tradition parallel/vector frontal solver used in ANSYS. Discussion of the applicability of iterative solvers as a general purpose solver will include the topics of robustness, as well as memory requirements and CPU performance. The FIDAP program is a widely used CFD code which uses iterative solvers routinely. A brief description of preconditioners used and some performance enhancements for CRAY parallel/vector systems is given. The solution of large-scale applications in structures and CFD includes examples from industry problems solved on CRAY systems.

Full Text Available We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM. Nonlinear stiffness matrices are constructed using Green-Lagrange strains, which are derived from infinitesimal strains by adding the nonlinear terms discarded from small deformations. We implemented a linear and a nonlinear finite element method with the same material properties to examine the differences between them. We verified our nonlinear formulation with different applications and achieved considerable speedups in solving the system of equations using our nonlinear FEM compared to a state-of-the-art nonlinear FEM.

Star centroiding accuracy decreases significantly when star sensor works under highly dynamic conditions or star images are corrupted by severe noise, reducing the output attitude precision. Herein, an adaptive iterationmethod is proposed to solve this problem. Firstly, initial star centroids are predicted by traditional method, and then based on initial reported star centroids and angular velocities of the star sensor, adaptive centroiding windows are generated to cover the star area and then an iterativemethod optimizing the location of centroiding window is used to obtain the final star spot extraction results. Simulation results shows that, compared with traditional star image restoration method and Iteratively Weighted Center of Gravity method, AWI algorithm maintains higher extraction accuracy when rotation velocities or noise level increases.

This article presents a numerical method for modeling nonlinear water waves based on the High Order Spectral (HOS) method proposed by Dommermuth and Yue and West et al., involving Taylor expansion of the Dirichlet problem and the Fast Fourier Transform (FFT) algorithm. The validation and efficiency of the numerical scheme is illustrated by a number of case studies on wave and wave train configuration including the evolution of fifth-order Stokes waves, wave dispersive focusing and the instability of Stokes wave with finite slope. The results agree well with those obtained by other studies.

Adaptive iterative reconstruction techniques (IRs) can decrease image noise in computed tomography (CT) and are expected to contribute to reduction of the radiation dose. To evaluate the performance of IRs, the conventional two-dimensional (2D) noise power spectrum (NPS) is widely used. However, when an IR provides an NPS value drop at all spatial frequency (which is similar to NPS changes by dose increase), the conventional method cannot evaluate the correct noise property because the conventional method does not correspond to the volume data natures of CT images. The purpose of our study was to develop a new method for NPS measurements that can be adapted to IRs. Our method utilized thick multi-planar reconstruction (MPR) images. The thick images are generally made by averaging CT volume data in a direction perpendicular to a MPR plane (e.g. z-direction for axial MPR plane). By using this averaging technique as a cutter for 3D-NPS, we can obtain adequate 2D-extracted NPS (eNPS) from 3D NPS. We applied this method to IR images generated with adaptive iterative dose reduction 3D (AIDR-3D, Toshiba) to investigate the validity of our method. A water phantom with 24 cm-diameters was scanned at 120 kV and 200 mAs with a 320-row CT (Acquilion One, Toshiba). From the results of study, the adequate thickness of MPR images for eNPS was more than 25.0 mm. Our new NPS measurement method utilizing thick MPR images was accurate and effective for evaluating noise reduction effects of IRs.

This paper presents an extension to classic component mode synthesis methods to compute the steady-state forced response of nonlinear and dissipative structures. The procedure makes use of the nonlinear complex modes of each substructure, computed by means of a modified harmonic balance method, in order to build a reduced-order model easily solved by standard iterative solvers. The proposed method is applied to a mistuned cyclic structure subjected to dry friction forces, and proves particularly suitable for the study of such systems with high modal density and non-conservative nonlinearities.

In seismic prospecting,fi eld conditions and other factors hamper the recording of the complete seismic wavefi eld; thus, data interpolation is critical in seismic data processing. Especially, in complex conditions, prestack missing data affect the subsequent high-precision data processing workfl ow. Compressive sensing is an effective strategy for seismic data interpolation by optimally representing the complex seismic wavefi eld and using fast and accurate iterative algorithms. The seislet transform is a sparse multiscale transform well suited for representing the seismic wavefield, as it can effectively compress seismic events. Furthermore, the Bregman iterative algorithm is an efficient algorithm for sparse representation in compressive sensing. Seismic data interpolation methods can be developed by combining seismic dynamic prediction, image transform, and compressive sensing. In this study, we link seismic data interpolation and constrained optimization. We selected the OC-seislet sparse transform to represent complex wavefields and used the Bregman iterationmethod to solve the hybrid norm inverse problem under the compressed sensing framework. In addition, we used an H-curve method to choose the threshold parameter in the Bregman iterationmethod. Thus, we achieved fast and accurate reconstruction of the seismic wavefi eld. Model andfi eld data tests demonstrate that the Bregman iterationmethod based on the H-curve norm in the sparse transform domain can effectively reconstruct missing complex wavefi eld data.

This work compares iterative 2D-3D Pose Estimation methods for use in real-time applications. The compared methods are available for public as C++ code. One method is part of the openCV library, namely POSIT. Because POSIT is not applicable for planar 3Dpoint congurations, we include the planar...

In this paper, variational iterationmethod and homotopy perturbation method are applied to different forms of diffusion equation. The diffusion equations have found wide applications in heat transfer problems, theory of consolidation and many other problems in engineering. The methods proposed t...

Full Text Available An approach based on homotopy iteration algorithm is proposed to identify the crack parameters in beam structures. In the forward problem, a fully open crack model with the composite element method is employed for the vibration analysis. The dynamic responses of the cracked beam in time domain are obtained from the Newmark direct integration method. In the inverse analysis, an identification approach based on homotopy iteration algorithm is studied to identify the location and the depth of a cracked beam. The identification equation is derived by minimizing the error between the calculated acceleration response and the simulated measured one. Newton iterativemethod with the homotopy equation is employed to track the correct path and improve the convergence of the crack parameters. Two numerical examples are conducted to illustrate the correctness and efficiency of the proposed method. And the effects of the influencing parameters, such as measurement time duration, measurement points, division of the homotopy parameter and measurement noise, are studied.

...) to object or shape reconstruction based on the singular sources method (or probe method) for the reconstruction of scatterers from the far-field pattern of scattered acoustic or electromagnetic waves...

A nonlinearmethod for locating defects in solid materials is discussed that is relevant to nonlinear modal tomography based on the signal cross-modulation. The scheme is illustrated by a theoretical model in which a thin plate or bar with a single crack is excited by a strong low-frequency wave and a high-frequency probing wave (ultrasound). A crack is considered as a small contact-type defect which does not perturb the modal structure of sound in linear approximation but creates combinational-frequency components whose amplitudes depend on their closeness to a resonance and crack position. Using different crack models, including the hysteretic ones, the nonlinear part of its volume variations under the given stress and then the combinational wave components in the bar can be determined. Evidently, their amplitude depends strongly on the crack position with respect to the peaks or nodes of the corresponding linear signals which can be used for localization of the crack position. Exciting the sample by sweeping ultrasound frequencies through several resonances (modes) reduces the ambiguity in the localization. Some aspects of inverse problem solution are also discussed, and preliminary experimental results are presented.

If you have ever given up on a nuclear criticality calculation and terminated it because it took so long to converge, you might find this thesis of interest. The author develops three methods for improving the fission source convergence in nuclear criticality calculations for physical systems with high dominance ratios for which convergence is slow. The Fission Matrix Acceleration Method and the Fission Diffusion Synthetic Acceleration (FDSA) Method are acceleration methods that speed fission source convergence for both Monte Carlo and deterministic methods. The third method is a hybrid Monte Carlo method that also converges for difficult problems where the unaccelerated Monte Carlo method fails. The author tested the feasibility of all three methods in a test bed consisting of idealized problems. He has successfully accelerated fission source convergence in both deterministic and Monte Carlo criticality calculations. By filtering statistical noise, he has incorporated deterministic attributes into the Monte Carlo calculations in order to speed their source convergence. He has used both the fission matrix and a diffusion approximation to perform unbiased accelerations. The Fission Matrix Acceleration method has been implemented in the production code MCNP and successfully applied to a real problem. When the unaccelerated calculations are unable to converge to the correct solution, they cannot be accelerated in an unbiased fashion. A Hybrid Monte Carlo method weds Monte Carlo and a modified diffusion calculation to overcome these deficiencies. The Hybrid method additionally possesses reduced statistical errors.

Full Text Available In this paper, a Second degree generalized Jacobi Iterationmethod for solving system of linear equations, $Ax=b$ and discuss about the optimal values $a_{1}$ and $b_{1}$ in terms of spectral radius about for the convergence of SDGJ method of $x^{(n+1}=b_{1}[D_{m}^{-1}(L_{m}+U_{m}x^{(n}+k_{1m}]-a_{1}x^{(n-1}.$ Few numerical examples are considered to show that the effective of the Second degree Generalized Jacobi Iterationmethod (SDGJ in comparison with FDJ, FDGJ, SDJ.

This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code's parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.

In Ujevic [A new iterativemethod for solving linear systems, Appl. Math. Comput. 179 (2006) 725-730], the author obtained a new iterativemethod for solving linear systems, which can be considered as a modification of the Gauss-Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujevic's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujevic.

Full Text Available Based on He’s variational iterationmethod idea, we modified the fractional variational iterationmethod and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS. The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iterationmethod is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iterationmethod in searching for approximate solutions of the Schrödinger equations.

To study calculating method of settlement on top of extra-long large-diameter pile, the relevant research results were summarized. The hyperbola model, a nonlinear load transfer function, was introduced to establish the basic differential equation with load transfer method. Assumed that the displacement of pile shaft was the high order power series of buried depth, through merging the same orthometric items and arranging the relevant coefficients, the solution which could take the nonlinear pile-soil interaction and stratum properties of soil into account was solved by power series. On the basis of the solution, by determining the load transfer depth with criterion of settlement on pile tip, the method by making boundary conditions compatible was advised to solve the load-settlement curve of pile. The relevant flow chart and mathematic expressions of boundary conditions were also listed. Lastly, the load transfer methods based on both two-broken-line model and hyperbola model were applied to analyzing a real project. The related coefficients of fitting curves by hyperbola were not less than 0.96, which shows that the hyperbola model is truthfulness, and is propitious to avoid personal error. The calculating value of load-settlement curve agrees well with the measured one, which indicates that it can be applied in engineering practice and making the theory that limits the design bearing capacity by settlement on pile top comes true.

The generalized least squares (LS) problem appears in many application areas. Here W is an m × m symmetric positive definite matrix and A is an m × n matrix with m≥n. Since the problem has many solutions in rank deficient case, some special preconditioned techniques are adapted to obtain the minimum 2-norm solution. A block SOR method and the preconditioned conjugate gradient (PCG) method are proposed here. Convergence and optimal relaxation parameter for the block SOR method are studied. An error bound for the PCG method is given. The comparison of these methods is investigated. Some remarks on the implementation of the methods and the operation cost are given as well.

A two-step iterativemethod and its accelerated version for approximating outer inverse A2 T,S of an arbitrary matrix A are proposed. A convergence theorem for its existence is established. The rigorous error bounds are derived. Numerical experiments involving singular square, rectangular, random matrices and a sparse matrix obtained by discretization of the Poisson’s equation are solved. Iterations count, computational time and the error bounds are used to measure the performance of our method. On comparing our results with those of other iterativemethods, it is seen that significantly better performance is achieved. Thus, enhanced speed and accuracy from the computational points of view has resulted for our methodology.

It was proved numerically that the Domain Decomposition Method (DDM) with one layer overlapping grids is identical to the block iterativemethod of linear algebra equations. The results obtained using DDM could be in reasonable aggeement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.

In this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite-element method. The proposed solution methods, based on a suitable approximation of the Schur-complement matrix, are shown t

In this paper we review the recent advances in three sub-areas of iterative learning control (ILC): 1) linear ILC for linear processes, 2) linear ILC for nonlinear processes which are global Lipschitz continuous (GLC), and 3) nonlinear ILC for general nonlinear processes. For linear processes, we focus on several basic configurations of linear ILC. For nonlinear processes with linear ILC, we concentrate on the design and transient analysis which were overlooked and missing for a long period. For general classes of nonlinear processes, we demonstrate nonlinear ILC methods based on Lyapunov theory, which is evolving into a new control paradigm.

Model identifications based on orbit tracking methods are here extended to stochastic differential equations. In the present approach, deterministic and statistical features are introduced via the time evolution of ensemble averages and variances. The aforementioned quantities are shown to follow deterministic equations, which are explicitly written within a linear as well as a weakly nonlinear approximation. Based on such equations and the observed time series, a cost function is defined. Its minimization by simulated annealing or backpropagation algorithms then yields a set of best-fit parameters. This procedure is successfully applied for various sampling time intervals, on a stochastic Lorenz system.

Full Text Available The Optimal Variational Method (OVM is introduced and applied for calculating approximate periodic solutions of “truly nonlinear oscillators”. The main advantage of this procedure consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. This approach does not depend upon any small or large parameters. A very good agreement was found between approximate and numerical solution, which proves that OVM is very efficient and accurate.

The variational iterationmethod is successfully extended to the case of solving fractional differential equations,and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.

Modifications of certain minimal iterationmethods for solving systems of linear algebraic equations are proposed and examined. The modified methods are shown to be superior to the original versions with respect to the round-off error accumulation, which makes them applicable to solving ill-conditioned problems. Numerical results demonstrating the efficiency of the proposed modifications are given.

This paper studies a delayed air-sea coupled oscillator describing the physical mechanism of El Ni(n)o Southern Oscillation.The approximate expansions of the delayed differential equation's solution are obtained successfully by the modified variational iterationmethod.The numerical results illustrate the effectiveness and correctness of the method by comparing with the exact solution of the reduced model.

Full Text Available The TTS iterativemethod is proposed to solve non-Hermitian positive definite linear systems and some convergence conditions are presented. Subsequently, these convergence conditions are applied to the ALUS method proposed by Xiang et al. in 2012 and comparison of some convergence theorems is made. Furthermore, an example is given to demonstrate the results obtained in this paper.

We have divided this book into five main chapters. Chapter 1 gives the motivation for this book and the use of templates. Chapter 2 describes stationary and nonstationary iterativemethods. In this chapter we present both historical development and state-of-the-art methods for solving some of the mo

Iterative and Sequential Bayesian Methods Zoi-Heleni Michalopoulou Department of Mathematical Sciences New Jersey Institute of Technology...exploiting (fully or partially) the physics of the propagation medium. Algorithms are designed for inversion via the extraction of features of the...statistical modeling. • Develop methods for passive localization and inversion of environmental parameters that select features of propagation that are

We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The general aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box (e.g. with commercial software). The proposed regularizing ensemble Kalman method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme (Hanke 1997 Inverse Problems 13 79-95) in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad hoc fixes, such as covariance localization, are implemented. The aim of this paper is to provide a detailed numerical investigation of the regularizing and convergence properties of the proposed regularizing ensemble Kalman scheme; the proof of these properties is an open problem. By means of numerical experiments, we investigate the conditions under which the proposed method inherits the regularizing properties of the LM scheme of (Hanke 1997 Inverse Problems 13 79-95) and is thus stable and suitable for its application in problems where the computation of the Fréchet derivative is not computationally feasible. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method

In the industrial process situation, principal component analysis (PCA) is a general method in data reconciliation.However, PCA sometime is unfeasible to nonlinear feature analysis and limited in application to nonlinear industrial process.Kernel PCA (KPCA) is extension of PCA and can be used for nonlinear feature analysis.A nonlinear data reconciliation method based on KPCA is proposed.The basic idea of this method is that firstly original data are mapped to high dimensional feature space by nonlinear function, and PCA is implemented in the feature space.Then nonlinear feature analysis is implemented and data are reconstructed by using the kernel.The data reconciliation method based on KPCA is applied to ternary distillation column.Simulation results show that this method can filter the noise in measurements of nonlinear process and reconciliated data can represent the true information of nonlinear process.

The Newton method has some shortcomings which includes computation of the Jacobian matrix which may be difficult or even impossible to compute and solving the Newton system in every iteration. Also, the common setback with some quasi-Newton methods is that they need to compute and store an n × n matrix at each iteration, this is computationally costly for large scale problems. To overcome such drawbacks, an improved Method for solving systems of nonlinear equations via PSB (Powell-Symmetric-Broyden) update is proposed. In the proposed method, the approximate Jacobian inverse Hk of PSB is updated and its efficiency has improved thereby require low memory storage, hence the main aim of this paper. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

Full Text Available Bioluminescence tomography (BLT has a great potential to provide a powerful tool for tumor detection, monitoring tumor therapy progress, and drug development; developing new reconstruction algorithms will advance the technique to practical applications. In the paper, we propose a BLT reconstruction algorithm by combining SP3 equations and Bregman iterationmethod to improve the quality of reconstructed sources. The numerical results for homogeneous and heterogeneous phantoms are very encouraging and give significant improvement over the algorithms without the use of SP3 equations and Bregman iterationmethod.

We investigate the suitability and effectiveness of iterativemethods for solving the weighted-least-square (WLS) state estimation problem on RISC, vector, and parallel processors. Several of the most popular iterativemethods are tested and evaluated. The best performing preconditioned conjugate gradient (PCG) is very well suited for vector and parallel processing as is demonstrated for the WLS state estimation of the IEEE standard test systems. A new sparse matrix format for the gain matrix improves vector performance of the PCG algorithm and makes it competitive to the direct solver. Internal parallelism in RISC processors, used in current multiprocessor systems, can be taken advantage of in an implementation of this algorithm.

Full Text Available The paper reviews prerequisites to creating a variety of the iterativemethods to solve a system of linear equations (SLE. It considers the splitting methods, variation-type methods, projection-type methods, and the methods of relaxation.A new iterativemethod based on mechanical analogy (the movement without resistance of a material point, that is connected by ideal elastically-linear constraints with unending guides defined by equations of solved SLE. The mechanical system has the unique position of stable equilibrium, the coordinates of which correspond to the solution of linear algebraic equation. The model of the mechanical system is a system of ordinary differential equations of the second order, integration of which allows you to define the point trajectory. In contrast to the classical methods of relaxation the proposed method does not ensure a trajectory passage through the equilibrium position. Thus the convergence of the method is achieved through the iterative stop of a material point at the moment it passes through the next (from the beginning of the given iteration minimum of potential energy. After that the next iteration (with changed initial coordinates starts.A resource-intensive process of numerical integration of differential equations in order to obtain a precise law of motion (at each iteration is replaced by defining its approximation. The coefficients of the approximating polynomial of the fourth order are calculated from the initial conditions, including higher-order derivatives. The resulting approximation enables you to evaluate the kinetic energy of a material point to calculate approximately the moment of time to reach the maximum kinetic energy (and minimum of the potential one, i.e. the end of the iteration.The software implementation is done. The problems with symmetric positive definite matrix, generated as a result of using finite element method, allowed us to examine a convergence rate of the proposed method

Full Text Available In order to enhance heat transfer between primary surface and the environment, radiating extended surfaces are commonly utilized. Especially in the case of large temperature differences, variable thermal conductivity has a strong effect on performance of such a surface. In this paper, variational iterationmethod is used to analyze convective straight and radial fins with temperature-dependent thermal conductivity. In order to show the efficiency of variational iterationmethod (VIM, the results obtained from VIM analysis are compared with previously obtained results using Adomian decomposition method (ADM and the results from finite element analysis. VIM produces analytical expressions for the solution of nonlinear differential equations. However, these expressions obtained from VIM must be tested with respect to the results obtained from a reliable numerical method or analytical solution. This work assures that VIM is a promising method for the analysis of convective straight and radial fin problems.

Recently Vilhena and Barichello proposed the LTSn method to solve, analytically, the Discrete Ordinates Problem (Sn problem) in transport theory. The main feature of this method consist in the application of the Laplace transform to the set of Sn equations and solve the resulting algebraic system for the transport flux. Barichello solve the linear system containing the parameter s applying the definition of matrix invertion exploiting the structure of the LTSn matrix. In this work, it is proposed a new scheme to invert the LTSn matrix, decomposing it in blocks and recursively inverting this blocks.

The finite element method is one of the most popular techniques for numerical solution of partial differential equations. The rapid performance increase of modern computer systems makes it possible to tackle increasingly more difficult finite-element models arising in engineering practice. However,

Full Text Available We discuss the convergence of GAOR method for linear systems with strictly diagonally dominant matrices. Moreover, we show that our results are better than ones of Darvishi and Hessari (2006, Tian et al. (2008 by using three numerical examples.

In this paper, we report results of the performance, convergence, and accuracy of a parallel GMRES solver for Boundary Element Methods. The solver uses a hierarchical approximate matrix-vector product based on a hybrid Barnes-Hut / Fast Multipole Method. We study the impact of various accuracy parameters on the convergence and show that with minimal loss in accuracy, our solver yields significant speedups. We demonstrate the excellent parallel efficiency and scalability of our solver. The combined speedups from approximation and parallelism represent an improvement of several orders in solution time. We also develop fast and paralellizable preconditioners for this problem. We report on the performance of an inner-outer scheme and a preconditioner based on truncated Green`s function. Experimental results on a 256 processor Cray T3D are presented.

A method and system for processing algae involves the use of an ionic liquid-containing clarified cell lysate to lyse algae cells. The resulting crude cell lysate may be clarified and subsequently used to lyse algae cells. The process may be repeated a number of times before a clarified lysate is separated into lipid and aqueous phases for further processing and/or purification of desired products.

In [4],Fletcher and Leyffer present a new method that solves nonlinear programming problems without a penalty function by SQP-Filter algorithm. It has attracted much attention due to its good numerical results. In this paper we propose a new SQP-Filter method which can overcome Maratos effect more effectively. We give stricter acceptant criteria when the iterative points are far from the optimal points and looser ones vice-versa. About this new method,the proof of global convergence is also presented under standard assumptions. Numerical results show that our method is efficient.

A system identification method for nonlinear systems with unknown structure is presented using short input-output data. The method simplifies the original NARMAX method. It introduces more general model structures for nonlinear systems. The group method of data handling (GMDH) method is employed to obtain the model terms and parameters. Effectiveness of the proposed method is illustrated by a typical nonlinear system with unknown structure and deficient input-output data.

Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group,an efficient numerical method is proposed for nonlinear dynamical systems.To improve computational efficiency,the integration step size can be adaptively controlled.Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system,the van der Pol system with strong stiffness,and the nonlinear Hamiltonian pendulum system.

Full Text Available We compare the computational performance of the Fast Marching Method, the Fast Sweeping Method and the Fast IterativeMethod to determine a numerical solution to the eikonal equation. We point out how the Fast IterativeMethod outperforms the other two thanks to its parallel processing capabilities.

The need for inclusion higher-order correlation effects for adequate description of the excitation energies of the DNA bases became clear in the last few years. In particular, we demonstrated that there is a sizable effect of triply excited configurations estimated in a non-iterative manner on the coupled-cluster excitation energies of the cytosine molecule in DNA environment. In this paper we discuss the accuracies of the non-iterativemethods for biologically relevant systems in realistic environment in comparison with interative formulations that explicitly include the effect of triply excited clusters.

This article presents an on-line tool (rttools.irap.omp.eu) and its accompanying software ressources for the numerical solution of basic radiation transfer out of local thermodynamic equilibrium (LTE). State-of-the-art stationary iterativemethods such as Accelerated $\\Lambda$-Iteration and Gauss-Seidel schemes, using a short characteristics-based formal solver are used. We also comment on typical numerical experiments associated to the basic non-LTE radiation problem. These ressources are intended for the largest use and benefit, in support to more classical radiation transfer lectures usually given at the Master level.

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) wide-angle (WA) beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator, which gives evanescent waves the desired damping. The resulting approach allows more accurate approximations to the true Helmholtz equation than the standard Padé approximant operators. Furthermore, a performance comparison of the traditional direct matrix inversion and this new iterative technique for WA-beam propagation method is reported. It is shown that complex Jacobi iteration is faster and better-suited for large problems or structures than direct matrix inversion.

Full Text Available We study displacement of a uniform elastic beam subject to various physically important boundary conditions. Using monotone methods, we discuss stability and instability of solutions. We present computations, which suggest efficiency of monotone methods for fourth order boundary value problems.

On the basis of a new dynamic linearization technology along the iteration axis, a dual-stage optimal iterative learning control is presented for nonlinear and non-affine discrete-time systems. Dual-stage indicates that two optimal learning stages are designed respectively to improve control input sequence and the learning gain iteratively. The main feature is that the controller design and convergence analysis only depend on the I/O data of the dynamical system. In other words, we can easily select the control parameters without knowing any other knowledge of the system. Simulation study illustrates the geometrical convergence of the presented method along the iteration axis, in which an example of freeway traffic iterative learning control is noteworthy for its intrinsic engineering importance.

We discuss a recent application of the asymptotic iterationmethod (AIM) to a perturbed Coulomb model. Contrary to what was argued before we show that the AIM converges and yields accurate energies for that model. We also consider alternative perturbation approaches and show that one of them is equivalent to that recently proposed by another author.

In this paper, we introduce a subclass of strictly quasi-nonexpansive operators which consists of well-known operators as paracontracting operators (e.g., strictly nonexpansive operators, metric projections, Newton and gradient operators), subgradient projections, a useful part of cutter operators, strictly relaxed cutter operators and locally strongly Féjer operators. The members of this subclass, which can be discontinuous, may be employed by fixed point iterationmethods; in particular, iterativemethods used in convex feasibility problems. The closedness of this subclass, with respect to composition and convex combination of operators, makes it useful and remarkable. Another advantage with members of this subclass is the possibility to adapt them to handle convex constraints. We give convergence result, under mild conditions, for a perturbation resilient iterativemethod which is based on an infinite pool of operators in this subclass. The perturbation resilient iterativemethods are relevant and important for their possible use in the framework of the recently developed superiorization methodology for constrained minimization problems. To assess the convergence result, the class of operators and the assumed conditions, we illustrate some extensions of existence research works and some new results.

Due to wide range of interest in use of bio-economic models to gain insight in to the scientific management of renewable resources like fisheries and forestry, variational iterationmethod (VIM) is employed to approximate the solution of the ratio-dependent predator-prey system with constant effort...

Full Text Available We study strong convergence of the sequence generated by implicit and explicit general iterativemethods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping Jφ, where φ is a gauge function on [0,∞. Our results improve and extend those announced by G. Marino and H.-K. Xu (2006 and many authors.

PageRank is one of the principle criteria according to which Google ranks Web pages. PageRank can be interpreted as a frequency of visiting a Web page by a random surfer, and thus it reflects the popularity of a Web page. Google computes the PageRank using the power iterationmethod, which requires

PageRank is one of the principle criteria according to which Google ranks Web pages. PageRank can be interpreted as a frequency of visiting a Web page by a random surfer and thus it reflects the popularity of a Web page. Google computes the PageRank using the power iterationmethod which requires ab

A fluid–structure interaction method combining a nonlinear finite element algorithm with a preconditioning finite volume method is proposed in this paper to simulate parachute tran-sient dynamics. This method uses a three-dimensional membrane–cable fabric model to represent a parachute system at a highly folded configuration. The large shape change during parachute infla-tion is computed by the nonlinear Newton–Raphson iteration and the linear system equation is solved by the generalized minimal residual (GMRES) method. A membrane wrinkling algorithm is also utilized to evaluate the special uniaxial tension state of membrane elements on the parachute canopy. In order to avoid large time expenses during structural nonlineariteration, the implicit Hil-ber–Hughes–Taylor (HHT) time integration method is employed. For the fluid dynamic simula-tions, the Roe and HLLC (Harten–Lax–van Leer contact) scheme has been modified and extended to compute flow problems at all speeds. The lower–upper symmetric Gauss–Seidel (LU-SGS) approximate factorization is applied to accelerate the numerical convergence speed. Finally, the test model of a highly folded C-9 parachute is simulated at a prescribed speed and the results show similar characteristics compared with experimental results and previous literature.

Full Text Available A fluid–structure interaction method combining a nonlinear finite element algorithm with a preconditioning finite volume method is proposed in this paper to simulate parachute transient dynamics. This method uses a three-dimensional membrane–cable fabric model to represent a parachute system at a highly folded configuration. The large shape change during parachute inflation is computed by the nonlinear Newton–Raphson iteration and the linear system equation is solved by the generalized minimal residual (GMRES method. A membrane wrinkling algorithm is also utilized to evaluate the special uniaxial tension state of membrane elements on the parachute canopy. In order to avoid large time expenses during structural nonlineariteration, the implicit Hilber–Hughes–Taylor (HHT time integration method is employed. For the fluid dynamic simulations, the Roe and HLLC (Harten–Lax–van Leer contact scheme has been modified and extended to compute flow problems at all speeds. The lower–upper symmetric Gauss–Seidel (LU-SGS approximate factorization is applied to accelerate the numerical convergence speed. Finally, the test model of a highly folded C-9 parachute is simulated at a prescribed speed and the results show similar characteristics compared with experimental results and previous literature.

This paper introduces a numerical method to estimate the region of attraction for polynomial nonlinear systems using sum of squares programming. This method computes a local Lyapunov function and an invariant set around a locally asymptotically stable equilibrium point. The invariant set is an estimation of the region of attraction for the equilibrium point. In order to enlarge the estimation, a subset of the invariant set defined by a shape factor is enlarged by solving a sum of squares optimization problem. In this paper, a new algorithm is proposed to select the shape factor based on the linearized dynamic model of the system. The shape factor is updated in each iteration using the computed local Lyapunov function from the previous iteration. The efficiency of the proposed method is shown by a few numerical examples.

A brief introduction to the basic principles of an eigen method used to obtain the optimum source location of lightning is presented. The location of the optimum source is obtained by using multiple direction finders (DF's) on a spherical surface. An improvement of this method, which takes the distance of source-DF's as a constant, is presented. It is pointed out that using a weight factor of signal strength is not the most ideal method because of the inexact inverse signal strength-distance relation and the inaccurate signal amplitude. An iterative calculation method is presented using the distance from the source to the DF as a weight factor. This improved method has higher accuracy and needs only a little more calculation time. Some computer simulations for a 4DF system are presented to show the improvement of location through use of the iterativemethod.

We present an extension of our GPGCD method, an iterativemethod for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.

We present an extension of our GPGCD method, an iterativemethod for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.

We present an extension of our GPGCD method, an iterativemethod for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.

Hydrologic frequency analysis plays an important role in coastal and ocean engineering for structural design and disaster prevention in coastal areas. This paper proposes a Nonlinear Least Squares Method (NLSM), which estimates the three unknown parameters of the Weibull distribution simultaneously by an iterationmethod. Statistical test shows that the NLSM fits each data sample well. The effects of different parameter-fitting methods, distribution models, and threshold values are also discussed in the statistical analysis of storm set-down elevation. The best-fitting probability distribution is given and the corresponding return values are estimated for engineering design.

Breast-cancer screening using microwave imaging is emerging as a new promising technique as a supplement to X-ray mammography. To create tomographic images from microwave measurements, it is necessary to solve a nonlinear inversion problem, for which an algorithm based on the iterative Gauss-Newton...... method has been developed at Dartmouth College. This algorithm determines the update values at each iteration by solving the set of normal equations of the problem using the Tikhonov algorithm. In this paper, a new algorithm for determining the iteration update values in the Gauss-Newton algorithm...... algorithm is compared to the Gauss-Newton algorithm with Tikhonov regularization and is shown to reconstruct images of similar quality using fewer iterations....

This paper develops and characterizes mixed direct-iterativemethods for boundary integral formulations of continuum dielectric solvation models. We give an example, the Ca{sup ++}{hor_ellipsis}Cl{sup {minus}} pair potential of mean force in aqueous solution, for which a direct solution at thermal accuracy is difficult and, thus for which mixed direct-iterativemethods seem necessary to obtain the required high resolution. For the simplest such formulations, Gauss-Seidel iteration diverges in rare cases. This difficulty is analyzed by obtaining the eigenvalues and the spectral radius of the non-symmetric iteration matrix. This establishes that those divergences are due to inaccuracies of the asymptotic approximations used in evaluation of the matrix elements corresponding to accidental close encounters of boundary elements on different atomic spheres. The spectral radii are then greater than one for those diverging cases. This problem is cured by checking for boundary element pairs closer than the typical spatial extent of the boundary elements and for those cases performing an ``in-line`` Monte Carlo integration to evaluate the required matrix elements. These difficulties are not expected and have not been observed for the thoroughly coarsened equations obtained when only a direct solution is sought. Finally, we give an example application of hybrid quantum-classical methods to deprotonation of orthosilicic acid in water.

Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.

According to various embodiments, a method is provided for determining aberration data for an optical system. The method comprises collecting a data signal, and generating a pre-transformation algorithm. The data is pre-transformed by multiplying the data with the pre-transformation algorithm. A discrete Fourier transform of the pre-transformed data is performed in an iterative loop. The method further comprises back-transforming the data to generate aberration data.

Using the cone and partial ordering theory and mixed monotone operator theory,the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.

For over 40 years, the Implicit Monte Carlo (IMC) method has been used to solve challenging problems in thermal radiative transfer. These problems typically contain regions that are optically thick and diffusive, as a consequence of the high degree of “pseudo-scattering” introduced to model the absorption and reemission of photons from a tightly-coupled, radiating material. IMC has several well-known features that could be improved: a) it can be prohibitively computationally expensive, b) it introduces statistical noise into the material and radiation temperatures, which may be problematic in multiphysics simulations, and c) under certain conditions, solutions can be nonphysical, in that they violate a maximum principle, where IMC-calculated temperatures can be greater than the maximum temperature used to drive the problem. We have developed a variant of IMC called iterative thermal emission IMC, which is designed to have a reduced parameter space in which the maximum principle is violated. ITE IMC is a more implicit version of IMC in that it uses the information obtained from a series of IMC photon histories to improve the estimate for the end of time step material temperature during a time step. A better estimate of the end of time step material temperature allows for a more implicit estimate of other temperature-dependent quantities: opacity, heat capacity, Fleck factor (probability that a photon absorbed during a time step is not reemitted) and the Planckian emission source. We have verified the ITE IMC method against 0-D and 1-D analytic solutions and problems from the literature. These results are compared with traditional IMC. We perform an infinite medium stability analysis of ITE IMC and show that it is slightly more numerically stable than traditional IMC. We find that significantly larger time steps can be used with ITE IMC without violating the maximum principle, especially in problems with non-linear material properties. The ITE IMC method does

Solving the static equilibrium position is one of the most important parts of dynamic coefficients calculation and further coupled calculation of rotor system. The main contribution of this study is testing the superlinear iteration convergence method-twofold secant method, for the determination of the static equilibrium position of journal bearing with finite length. Essentially, the Reynolds equation for stable motion is solved by the finite difference method and the inner pressure is obtained by the successive over-relaxation iterativemethod reinforced by the compound Simpson quadrature formula. The accuracy and efficiency of the twofold secant method are higher in comparison with the secant method and dichotomy. The total number of iterative steps required for the twofold secant method are about one-third of the secant method and less than one-eighth of dichotomy for the same equilibrium position. The calculations for equilibrium position and pressure distribution for different bearing length, clearance and rotating speed were done. In the results, the eccentricity presents linear inverse proportional relationship to the attitude angle. The influence of the bearing length, clearance and bearing radius on the load-carrying capacity was also investigated. The results illustrate that larger bearing length, larger radius and smaller clearance are good for the load-carrying capacity of journal bearing. The application of the twofold secant method can greatly reduce the computational time for calculation of the dynamic coefficients and dynamic characteristics of rotor-bearing system with a journal bearing of finite length.

Solving the static equilibrium position is one of the most important parts of dynamic coefficients calculation and further coupled calculation of rotor system. The main contribution of this study is testing the superlinear iteration convergence method-twofold secant method, for the determination of the static equilibrium position of journal bearing with finite length. Essentially, the Reynolds equation for stable motion is solved by the finite difference method and the inner pressure is obtained by the successive over-relaxation iterativemethod reinforced by the compound Simpson quadrature formula. The accuracy and efficiency of the twofold secant method are higher in comparison with the secant method and dichotomy. The total number of iterative steps required for the twofold secant method are about one-third of the secant method and less than one-eighth of dichotomy for the same equilibrium position. The calculations for equilibrium position and pressure distribution for different bearing length, clearance and rotating speed were done. In the results, the eccentricity presents linear inverse proportional relationship to the attitude angle. The influence of the bearing length, clearance and bearing radius on the load-carrying capacity was also investigated. The results illustrate that larger bearing length, larger radius and smaller clearance are good for the load-carrying capacity of journal bearing. The application of the twofold secant method can greatly reduce the computational time for calculation of the dynamic coefficients and dynamic characteristics of rotor-bearing system with a journal bearing of finite length.

Multigrid and iterativemethods are used to reduce the solution time of the matrix equations which arise from the finite element (FE) discretisation of the time-independent equations of motion of the incompressible fluid in turbulent motion. Incompressible flow is solved by using the method of reduce interpolation for the pressure to satisfy the Brezzi-Babuska condition. The k-l model is used to complete the turbulence closure problem. The non-symmetric iterative matrix methods examined are the methods of least squares conjugate gradient (LSCG), biconjugate gradient (BCG), conjugate gradient squared (CGS), and the biconjugate gradient squared stabilised (BCGSTAB). The multigrid algorithm applied is based on the FAS algorithm of Brandt, and uses two and three levels of grids with a V-cycling schedule. These methods are all compared to the non-symmetric frontal solver. Copyright

A method is given for the shape, permittivity and conductivity reconstruction of lossy dielectric objects buried under rough surfaces using the Distorted Born IterativeMethod (DBIM). The method is based on the refreshing of the Green's function of the two-part space media with rough interface by updating the complex permittivity of the reconstruction domain at each iteration step. The scattered field data are measured at multiple locations for multiple transmitters operating at a single frequency where both transmitters and receivers are located above the rough surface interface. The Green's function of the problem is obtained by using the buried object approach (BOA) method where the fluctuations of the rough surface from the flat one are assumed to be buried objects in a two-part space with planar interface. The performance of the method is tested by some numerical applications and satisfactory results are obtained.

μ synthesis method for robust control of uncertain nonlinear systems is propored, which is based on feedback linearization. First, nonlinear systems are linearized as controllable linear systems by I/O linearization,such that uncertain nonlinear systems are expressed as the linear fractional transformations (LFTs) on the generalized linearized plants and uncertainty.Then,linear robust controllers are obtained for the LFTs usingμsynthesis method based on H∞ optimization.Finally,the nonlinear robust controllers are constructed by combining the linear robust controllers and the nonlinear feedback.An example is given to illustrate the design.

Full Text Available This paper presents a variant of the iterative shrinkage-thresholding (IST algorithm, called backtracking-based adaptive IST (BAIST, for image compressive sensing (CS reconstruction. For increasing iterations, IST usually yields a smoothing of the solution and runs into prematurity. To add back more details, the BAIST method backtracks to the previous noisy image using L2 norm minimization, i.e., minimizing the Euclidean distance between the current solution and the previous ones. Through this modification, the BAIST method achieves superior performance while maintaining the low complexity of IST-type methods. Also, BAIST takes a nonlocal regularization with an adaptive regularizor to automatically detect the sparsity level of an image. Experimental results show that our algorithm outperforms the original IST method and several excellent CS techniques.

This paper describes new extensions to the previously published multivariate alteration detection (MAD) method for change detection in bi-temporal, multi- and hypervariate data such as remote sensing imagery. Much like boosting methods often applied in data mining work, the iteratively reweighted...... an agricultural region in Kenya, and hyperspectral airborne HyMap data from a small rural area in southeastern Germany are given. The latter case demonstrates the need for regularization....... (IR) MAD method in a series of iterations places increasing focus on “difficult” observations, here observations whose change status over time is uncertain. The MAD method is based on the established technique of canonical correlation analysis: for the multivariate data acquired at two points in time...

An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems withstrongly discontinuous solutions, conormal derivatives, and coefficients. This algorithmiteratively solves small problems for each single phase with good accuracy and exchangeinformation at the interface to advance the iteration until convergence, following the ideaof Schwarz Alternating Methods. Error estimates are derived to show that this algorithmalways converges provided that relaxation parameters are suitably chosen. Numeric experiments with matching and non-matching grids at the interface from different phases areperformed to show the accuracy of the method for capturing discontinuities in the solutionsand coefficients. In contrast to standard numerical methods, the accuracy of our methoddoes not seem to deteriorate as the coefficient discontinuity increases.

Full Text Available In this paper, we apply the modified variational iterationmethod to a generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV equation. The numerical solutions of the initial value problem of the generalized Hirota-Satsuma coupled KdV equation are provided. Numerical results are given to show the efficiency of the modified variational iterationmethod.

Our proposed method of random phase-free holography using virtual convergence light can obtain large reconstructed images exceeding the size of the hologram, without the assistance of random phase. The reconstructed images have low-speckle noise in the amplitude and phase-only holograms (kinoforms); however, in low-resolution holograms, we obtain a degraded image quality compared to the original image. We propose an iterative random phase-free method with virtual convergence light to address this problem.

We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).

The best linear unbiased estimator (BLUE) is a popular statistical method adopted to combine multiple measurements of the same observable, taking into account individual uncertainties and their correlation. The method is unbiased by construction if the true uncertainties and their correlation are known, but it may exhibit a bias if uncertainty estimates are used in place of the true ones, in particular if those uncertainties depend on the true value of the measured quantity. This is the case for instance when contributions to the total uncertainty are known as relative uncertainties. In those cases, an iterative application of the BLUE method may reduce the bias of the combined measurement. The impact of the iterative approach compared to the standard BLUE application is studied for a wide range of possible values of uncertainties and their correlation in the case of the combination of two measurements.

We develop a novel iterativemethod to accurately measure electron beam shape (current density distribution) and monotonic material response as a function of position. A common method is to scan an electron beam across a knife edge along many angles to give an approximate measure of the beam profile, however such scans are not easy to obtain in all systems. The present work uses only an electron beam and multiple exposed regions of a thin film of photoresist to measure the complete beam profile for any beam shape, where the material response is characterized externally. This simplifies the setup of new experimental tools. We solve for self-consistent photoresist thickness loss response to dose and the electron beam profile simultaneously by optimizing a novel functional iteratively. We also show the successful implementation of the method in a real world data set corrupted by noise and other experimental variabilities.

Full Text Available Optical molecular imaging is a promising technique and has been widely used in physiology, and pathology at cellular and molecular levels, which includes different modalities such as bioluminescence tomography, fluorescence molecular tomography and Cerenkov luminescence tomography. The inverse problem is ill-posed for the above modalities, which cause a nonunique solution. In this paper, we propose an effective reconstruction method based on the linearized Bregman iterative algorithm with sparse regularization (LBSR for reconstruction. Considering the sparsity characteristics of the reconstructed sources, the sparsity can be regarded as a kind of a priori information and sparse regularization is incorporated, which can accurately locate the position of the source. The linearized Bregman iterationmethod is exploited to minimize the sparse regularization problem so as to further achieve fast and accurate reconstruction results. Experimental results in a numerical simulation and in vivo mouse demonstrate the effectiveness and potential of the proposed method.

研究了计算大型稀疏对称矩阵的若干个最大或最小特征值的问题,首先引入了求解大型对称特征值问题的预处理子空间迭代法和Chebyshev迭代法,并对其作了理论分析.为了加速顶处理子空间迭代法的收敛性,笔者采用组合Chebyshev迭代法和预处理子空间选代法,提出了计算大型对称稀疏矩阵的几个最大或最小特征值的Chebyshev预处理子空间迭代法.数值结果表明,该方法比预处理子空间方法优越.%The problem of computing a few of the largest (or smallest) eigenvalues of a large symmetric sparse matrix is dealt with. This paper considers the preconditioning subspace iterationmethod and the Chebyshev iteration, and analyzes them. In order to accelerate the convergence rate of the preconditioning subspace iterationmethod,a new method, i. e. Chebyshev -PSI(the preconditioning subspace iteration) method, is presented for computing the extreme eigenvalues of a large symmetric sparse matrix. The new method combines the Chebyshev iteration with the PSI method. Numerical experiments show that the Chebyshev - PS1 metod is very effective for computing the extreme eigenvalues of a large symmetric sparse matrix.

This study addresses the mitigation of a nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog's equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments.

This thesis addresses a posteriori error estimation for finite element methods and iterative linear solvers. Adaptive finite element methods have gained a lot of popularity over the last decades due to their ability to produce accurate results with limited computer power. In these methods a posteriori error estimates play an essential role. Not only do they give information about how large the total error is, they also indicate which parts of the computational domain should be given a more sophisticated treatment in order to reduce the error. A posteriori error estimates are traditionally aimed at estimating the global error, but more recently so called goal oriented error estimators have been shown a lot of interest. The name reflects the fact that they estimate the error in user-defined local quantities. In this thesis the main focus is on global error estimators for highly stretched grids and goal oriented error estimators for flow problems on regular grids. Numerical methods for partial differential equations, such as finite element methods and other similar techniques, typically result in a linear system of equations that needs to be solved. Usually such systems are solved using some iterative procedure which due to a finite number of iterations introduces an additional error. Most such algorithms apply the residual in the stopping criterion, whereas the control of the actual error may be rather poor. A secondary focus in this thesis is on estimating the errors that are introduced during this last part of the solution procedure. The thesis contains new theoretical results regarding the behaviour of some well known, and a few new, a posteriori error estimators for finite element methods on anisotropic grids. Further, a goal oriented strategy for the computation of forces in flow problems is devised and investigated. Finally, an approach for estimating the actual errors associated with the iterative solution of linear systems of equations is suggested. (author)

Many diverse applied methods of investigating oscillations of nonlinear systems often in different mathematical formulations and outwardly not...parameter classical methods and the methods of investigating nonlinear systems of automatic control based on the so-called filter hypothesis, and to

This paper proposes a new method to numerically obtain Floquet multipliers which characterize stability of periodic orbits of ordinary differential equations. For sufficiently smooth periodic orbits, we can compute Floquet multipliers using some standard numerical methods with enough accuracy. However, it has been reported that these methods may produce incorrect results under some conditions. In this work, we propose a new iterativemethod to compute Floquet multipliers using eigenvectors of matrix solutions of the variational equations. Numerical examples show effectiveness of the proposed method.

Full Text Available In this research, an efficient technique of computation considered as a modified decomposition method was proposed and then successfully applied for solving the nonlinear problem of the two dimensional flow of an incompressible viscous fluid between nonparallel plane walls. In fact this method gives the nonlinear term Nu and the solution of the studied problem as a power series. The proposed iterative procedure gives on the one hand a computationally efficient formulation with an acceleration of convergence rate and on the other hand finds the solution without any discretization, linearization or restrictive assumptions. The comparison of our results with those of numerical treatment and other earlier works shows clearly the higher accuracy and efficiency of the used Modified Decomposition Method.

Full Text Available Exposure to radiation is one of the main sources of risk to staff employed in reactor facilities. The staff of a tokamak is exposed to a wide range of neutrons and photons around the tokamak hall. The International Thermonuclear Experimental Reactor (ITER is a nuclear fusion engineering project and the most advanced experimental tokamak in the world. From the radiobiological point of view, ITER dose rates assessment is particularly important. The aim of this study is the assessment of the amount of radiation in ITER during its normal operation in a radial direction from the plasma chamber to the tokamak hall. To achieve this goal, the ITER system and its components were simulated by the Monte Carlo method using the MCNPX 2.6.0 code. Furthermore, the equivalent dose rates of some radiosensitive organs of the human body were calculated by using the medical internal radiation dose phantom. Our study is based on the deuterium-tritium plasma burning by 14.1 MeV neutron production and also photon radiation due to neutron activation. As our results show, the total equivalent dose rate on the outside of the bioshield wall of the tokamak hall is about 1 mSv per year, which is less than the annual occupational dose rate limit during the normal operation of ITER. Also, equivalent dose rates of radiosensitive organs have shown that the maximum dose rate belongs to the kidney. The data may help calculate how long the staff can stay in such an environment, before the equivalent dose rates reach the whole-body dose limits.

The problem of atmospheric tomography arises in ground-based telescope imaging with adaptive optics (AO), where one aims to compensate in real-time for the rapidly changing optical distortions in the atmosphere. Many of these systems depend on a sufficient reconstruction of the turbulence profiles in order to obtain a good correction. Due to steadily growing telescope sizes, there is a strong increase in the computational load for atmospheric reconstruction with current methods, first and foremost the MVM. In this paper we present and compare three novel iterative reconstruction methods. The first iterative approach is the Finite Element- Wavelet Hybrid Algorithm (FEWHA), which combines wavelet-based techniques and conjugate gradient schemes to efficiently and accurately tackle the problem of atmospheric reconstruction. The method is extremely fast, highly flexible and yields superior quality. Another novel iterative reconstruction algorithm is the three step approach which decouples the problem in the reconstruction of the incoming wavefronts, the reconstruction of the turbulent layers (atmospheric tomography) and the computation of the best mirror correction (fitting step). For the atmospheric tomography problem within the three step approach, the Kaczmarz algorithm and the Gradient-based method have been developed. We present a detailed comparison of our reconstructors both in terms of quality and speed performance in the context of a Multi-Object Adaptive Optics (MOAO) system for the E-ELT setting on OCTOPUS, the ESO end-to-end simulation tool.

Given a set of alternatives to be ranked and some pairwise comparison values, ranking can be posed as a least squares computation on a graph. This was first used by Leake for ranking football teams. The residual can be further analyzed to find inconsistencies in the given data, and this leads to a second least squares problem. This whole process was formulated recently by Jiang et al. as a Hodge decomposition of the edge values. Recently, Koutis et al., showed that linear systems involving symmetric diagonally dominant (SDD) matrices can be solved in time approaching optimality. By using Hodge 0-Laplacian and 2-Laplacian, we give various results on when the normal equations for ranking are SDD and when iterative Krylov methods should be used. We also give iteration bounds for conjugate gradient method for these problems.

Multi-qubit entanglement states are the key resources for various multipartite quantum communication tasks. For a class of generalized three-qubit quantum entanglement, W-like state, we demonstrate that the weak measurement and the reversal measurement are capable of suppressing the amplitude damping decoherence by reducing the initial damping factor into a smaller equivalent damping factor. Furthermore, we propose an iterationmethod in the weak measurement and the reversal measurement to enhance the success probability of the total measurements. Finally, we discuss how the number of the iterations influences the overall effect of decoherence suppression, and find that the “half iteration” method is a better option that has more practical value. Project supported by the National Natural Science Foundation of China (Grant No. 61172071), the International Scientific Cooperation Program of Shaanxi Province, China (Grant No. 2015KW-013), and the Scientific Research Program Funded by Shaanxi Provincial Education Department, China (Grant No. 16JK1711).

Full Text Available In this paper, a new algorithm is proposed for solving the path planning problem of mobile robots. The algorithm is based on Artificial Potential Field (APF methods that have been widely used for path planning related problems for more than two decades. While keeping the simplicity of traditional APF methods, our algorithm is built upon new potential functions based on the distances from obstacles, destination point and start point. The algorithm uses the potential field values iteratively to find the optimum points in the workspace in order to form the path from start to destination. The number of iterations depends on the size and shape of the workspace. The performance of the proposed algorithm is tested by conducting simulation experiments.

The measurement of material non-linearity using ultrasound is an attractive concept, offering the potential to detect fatigue damage earlier than is possible with conventional techniques. Despite this advantage and much work in the field the currently developed approaches are primarily limited to the lab environment. This is due to the difficulty in separating the material nonlinearity from that generated by equipment. This paper reports on an approach that eliminates this problem. When two shear waves interact a third wave is generated due to the material nonlinearity. This paper shows how this interaction can be used to measure material properties in damaged specimens. It goes on to show that this approach can be used to make measurements of material non-linearity both across a specimen.

Full Text Available This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE involving Jumarie’s modification of Riemann-Liouville derivative by the fractional variational iterationmethod (FVIM. FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs.

In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been proposed. This paper gives a survey of some of these recent developments.

Full Text Available We are concerned here with singular partial differential equations of fractional order (FSPDEs. The variational iterationmethod (VIM is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

Among many molecular imaging modalities, Bioluminescence tomography (BLT) is an important optical molecular imaging modality. Due to its unique advantages in specificity, sensitivity, cost-effectiveness and low background noise, BLT is widely studied for live small animal imaging. Since only the photon distribution over the surface is measurable and the photo propagation with biological tissue is highly diffusive, BLT is often an ill-posed problem and may bear multiple solutions and aberrant reconstruction in the presence of measurement noise and optical parameter mismatches. For many BLT practical applications, such as early detection of tumors, the volumes of the light sources are very small compared with the whole body. Therefore, the L1-norm sparsity regularization has been used to take advantage of the sparsity prior knowledge and alleviate the ill-posedness of the problem. Iterative shrinkage (IST) algorithm is an important research achievement in a field of compressed sensing and widely applied in sparse signal reconstruction. However, the convergence rate of IST algorithm depends heavily on the linear operator. When the problem is ill-posed, it becomes very slow. In this paper, we present a sparsity regularization reconstruction method for BLT based on the two-step iterated shrinkage approach. By employing Two-step strategy of iterative reweighted shrinkage (IRS) to improve IST, the proposed method shows faster convergence rate and better adaptability for BLT. The simulation experiments with mouse atlas were conducted to evaluate the performance of proposed method. By contrast, the proposed method can obtain the stable and comparable reconstruction solution with less number of iterations.

Full Text Available The paper uses the Local fractional variational IterationMethod for solving the second kind Volterra integro-differential equations within the local fractional integral operators. The analytical solutions within the non-differential terms are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the integral equations.

Complete complementary sequences are widely used in spectrum spread communications because of their ideal correlation functions. A previous method generates complete complementary sequences with lengths of NnN (n,N∈Z+). This paper presents a new iterativemethod to construct complete complementary sequences with lengths of 2mN (m,N∈Z+). The analysis proves that this method can produce many sequence sets that do not appear in sequence sets generated by the former method, especially shorter sequence sets. The result will certainly increase the application of complete complementary sequences in communication engineering and related fields.

In this paper we prove that the classical Gauss-Seidel type iterativemethods used for the solution of the reduced normal equations occurring in overlapping reduction methods of astrometry do not always converge. We exhibit examples of divergence. We then analyze an alternative algorithm proposed by Wang (1985). We prove the consistency of this algorithm and verify that it can be convergent while the Gauss-Seidel method is divergent. We conjecture the convergence of Wang method for the solution of astrometric problems using overlap techniques.

Highlights: • An in operation force measurement system for the ITER EU HCPB TBM has been developed. • The force reconstruction methods are based on strain measurements on the attachment system. • An experimental setup and a corresponding mock-up have been built. • A set of test cases representing ITER relevant excitations has been used for validation. • The influence of modeling errors on the force reconstruction has been investigated. - Abstract: In order to reconstruct forces on the test blanket modules in ITER, two force reconstruction methods, the augmented Kalman filter and a model predictive controller, have been selected and developed to estimate the forces based on strain measurements on the attachment system. A dedicated experimental setup with a corresponding mock-up has been designed and built to validate these methods. A set of test cases has been defined to represent possible excitation of the system. It has been shown that the errors in the estimated forces mainly depend on the accuracy of the identified model used by the algorithms. Furthermore, it has been found that a minimum of 10 strain gauges is necessary to allow for a low error in the reconstructed forces.

Full Text Available The conjugate gradient method is an efficient method for solving large-scale nonlinear optimization problems. In this paper, we propose a nonlinear conjugate gradient method which can be considered as a hybrid of DL and WYL conjugate gradient methods. The given method possesses the sufficient descent condition under the Wolfe-Powell line search and is globally convergent for general functions. Our numerical results show that the proposed method is very robust and efficient for the test problems.

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.

prediction of free-surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow......Robust computational procedures for the solution of non-hydrostatic, free surface, irrotational and inviscid free-surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable...... models. Our study is particularly relevant for fast and efficient simulation of non-breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization...

We present for the first time an efficient iterativemethod to directly solve the four-component Dirac-Kohn-Sham (DKS) density functional theory. Due to the existence of the negative energy continuum in the DKS operator, the existing iterative techniques for solving the Kohn-Sham systems cannot be efficiently applied to solve the DKS systems. The key component of our method is a novel filtering step (F) which acts as a preconditioner in the framework of the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The resulting method, dubbed the LOBPCG-F method, is able to compute the desired eigenvalues and eigenvectors in the positive energy band without computing any state in the negative energy band. The LOBPCG-F method introduces mild extra cost compared to the standard LOBPCG method and can be easily implemented. We demonstrate our method in the pseudopotential framework with a planewave basis set which naturally satisfies the kinetic balance prescription. Numerical results for Pt$_{2}$, Au$_{2}$, TlF, and Bi$_{2}$Se$_{3}$ indicate that the LOBPCG-F method is a robust and efficient method for investigating the relativistic effect in systems containing heavy elements.

A 3-D time-domain seakeeping analysis tool has been newly developed by using a higher-order boundary element method with the Rankine source as the kernel function. An iterative time-marching scheme for updating both kinematic and dynamic free-surface boundary conditions is adopted for achieving numerical accuracy and stability. A rectangular computational domain moving with the mean speed of ship is introduced. A damping beach at the outer portion of the truncated free surface is installed for satisfying the radiation condition. After numerical convergence checked, the diffraction unsteady problem of a Wigley hull traveling with a constant forward speed in waves is studied. Extensive results including wave exciting forces, wave patterns and pressure distributions on the hull are presented to validate the efficiency and accuracy of the proposed 3-D time-domain iterative Rankine BEM approach. Computed results are compared to be in good agreement with the corresponding experimental data and other published numerical solutions.

Full Text Available A new iterativemethod for polynomial root-finding based on the development of two novel recursive functions is proposed. In addition, the concept of polynomial pivots associated with these functions is introduced. The pivots present the property of lying close to some of the roots under certain conditions; this closeness leads us to propose them as efficient starting points for the proposed iterative sequences. Conditions for local convergence are studied demonstrating that the new recursive sequences converge with linear velocity. Furthermore, an a priori checkable global convergence test inside pivots-centered balls is proposed. In order to accelerate the convergence from linear to quadratic velocity, new recursive functions together with their associated sequences are constructed. Both the recursive functions (linear and the corrected (quadratic convergence are validated with two nontrivial numerical examples. In them, the efficiency of the pivots as starting points, the quadratic convergence of the proposed functions, and the validity of the theoretical results are visualized.

Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, $p$. We take advantage of a unique property of Krylov iterations that allow lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing $p$. Via extensive numerical tests, we show that the relaxed Krylov iterations converge with speed-ups of between 2x and 4x for Laplace problems and between 3.5x and 4.5x for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a quad-core CPU.

To ensure the system run under working order, detection and diagnosis of faults play an important role in industrial process. This paper proposed a nonlinear fault diagnosis method based on kernel principal component analysis (KPCA). In proposed method, using essential information of nonlinear system extracted by KPCA, we constructed KPCA model of nonlinear system under normal working condition. Then new data were projected onto the KPCA model. When new data are incompatible with the KPCA model, it can be concluded that the nonlinear system isout of normal working condition. Proposed method was applied to fault diagnosison rolling bearings. Simulation results show proposed method provides an effective method for fault detection and diagnosis of nonlinear system.

Full Text Available This paper discusses the inverse problem of determining an unknown source in a second order differential equation from measured final data. This problem is ill-posed; that is, the solution (if it exists does not depend continuously on the data. In order to solve the considered problem, an iterativemethod is proposed. Using this method a regularized solution is constructed and an a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

In this article, we propose a new approach for solving fractional partial differential equations with variable coefficients, which is very effective and can also be applied to other types of differential equations. The main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations. The fractional derivatives are described based on the Caputo sense. Our method contains an iterative formula that can provide rapidly convergent successive approximations of the exact solution if such a closed form solution exists. Several examples are given, and the numerical results are shown to demonstrate the efficiency of the newly proposed method.

Threshold Segmentation is a kind of important image segmentation method and one of the important preconditioning steps of image detection and recognition, and it has very broad application during the research scopes of the computer vision. According to the internal relation between segment image and original image, a tomato image automatic optimization segmentation method (MI-OPT) which mutual information associate with optimum threshold iteration was presented. Simulation results show that this method has a better image segmentation effect on the tomato images of mature period and little background color difference or different color.

This paper shows how to generalize to non-LTE polarization transfer some operator splitting methods that were originally developed for solving unpolarized transfer problems. These are the Jacobi-based accelerated Λ-iteration (ALI) method of Olson, Auer, & Buchler and the iterative schemes based on Gauss-Seidel and successive overrelaxation (SOR) iteration of Trujillo Bueno and Fabiani Bendicho. The theoretical framework chosen for the formulation of polarization transfer problems is the quantum electrodynamics (QED) theory of Landi Degl'Innocenti, which specifies the excitation state of the atoms in terms of the irreducible tensor components of the atomic density matrix. This first paper establishes the grounds of our numerical approach to non-LTE polarization transfer by concentrating on the standard case of scattering line polarization in a gas of two-level atoms, including the Hanle effect due to a weak microturbulent and isotropic magnetic field. We begin demonstrating that the well-known Λ-iterationmethod leads to the self-consistent solution of this type of problem if one initializes using the ``exact'' solution corresponding to the unpolarized case. We show then how the above-mentioned splitting methods can be easily derived from this simple Λ-iteration scheme. We show that our SOR method is 10 times faster than the Jacobi-based ALI method, while our implementation of the Gauss-Seidel method is 4 times faster. These iterative schemes lead to the self-consistent solution independently of the chosen initialization. The convergence rate of these iterativemethods is very high; they do not require either the construction or the inversion of any matrix, and the computing time per iteration is similar to that of the Λ-iterationmethod.

Full Text Available We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

Floquet eigenanalysis requires a few dominant eigenvalues of the Floquet transition matrix (FTM). Although the QR method is used almost exclusively, it is expensive for such partial eigenanalysis; the operation counts and, thereby, the approximate machine-time grow cubically with the matrix order. Accordingly, for Floquet eigenanalysis, the Arnold-Saad method, a subspace iterationmethod, is investigated as an alternative to the QR method. The two methods are compared for machine-time efficiency and the residual errors of the corresponding eigenpairs. The Arnolds-Saad method takes much less machine-time than the QR method with comparable computational reliability and offers promise fpr large-scale Floquet eigenanalysis.

A dynamic decoupling nonlinear control method for MIMO system is presented in this paper. The dynamic inversion method is used to decouple the multivariable system. The nonlinear control method is used to overcome the poor decoupling effect when the system model is inaccurate. The nonlinear control method has correcting function and is expressed in analytic form, it is easy to adjust the parameters of the controller and optimize the design of the control system. The method is used to design vertical transition mode of active control aircraft for gust alleviation. Simulation results show that the designed vertical transition mode improves the gust alleviation effect about 34% comparing with the normal aircraft.

Full Text Available In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to 814≈1.682${8^{{\\textstyle{1 \\over 4}}}} \\approx 1.682$. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

This Lecture Note deals with asymptotic properties, i.e. weak and strong consistency and asymptotic normality, of parameter estimators of nonlinear regression models and nonlinear structural equations under various assumptions on the distribution of the data. The estimation methods involved are nonlinear least squares estimation (NLLSE), nonlinear robust M-estimation (NLRME) and non­ linear weighted robust M-estimation (NLWRME) for the regression case and nonlinear two-stage least squares estimation (NL2SLSE) and a new method called minimum information estimation (MIE) for the case of structural equations. The asymptotic properties of the NLLSE and the two robust M-estimation methods are derived from further elaborations of results of Jennrich. Special attention is payed to the comparison of the asymptotic efficiency of NLLSE and NLRME. It is shown that if the tails of the error distribution are fatter than those of the normal distribution NLRME is more efficient than NLLSE. The NLWRME method is appropriate ...

The properly-posedness of the nonlinear fixed gravimetric boundary value problem is shown with the help of nonlinear functional analysis and a new iterativemethod to solve the problem is also given, where each step of the iterative program is reduced to solving one and the same kind of oblique derivative boundary value problem with the same type. Furthermore, the convergence of the iterative program is proved with Schauder estimate of elliptic differential equation.

element is proposed to solve the safety factor of local discontinuous rock mass. Slope system is divided into several continuous bodies and local discontinuous interface boundaries. Each block is treated as a partition of the system and contacted by discontinuous joints. The displacements of blocks are chosen as basic variables and the rigid displacements in the centroid of blocks are chosen as motion variables. The contact forces on interface boundaries and the rigid displacements to the centroid of each body are chosen as mixed variables and solved iteratively using the interface boundary equations. Flexibility matrix is formed through PFE according to the contact states of nodal pairs and spring flexibility is used to reflect the influence of weak structural plane so that nonlineariteration is only limited to the possible contact region. With cohesion and friction coefficient reduced gradually, the states of all nodal pairs at the open or slip state for the first time are regarded as failure criterion, which can decrease the effect of subjectivity in determining safety factor. Examples are used to verify the validity of the proposed method.

Motivated by recent developments in perturbative calculations of the nonlinear evolution of large-scale structure, we present an iterative algorithm to reconstruct the initial conditions in a given volume starting from the dark matter distribution in real space. In our algorithm, objects are first moved back iteratively along estimated potential gradients, with a progressively reduced smoothing scale, until a nearly uniform catalog is obtained. The linear initial density is then estimated as the divergence of the cumulative displacement, with an optional second-order correction. This algorithm should undo nonlinear effects up to one-loop order, including the higher-order infrared resummation piece. We test the method using dark matter simulations in real space. At redshift z =0 , we find that after eight iterations the reconstructed density is more than 95% correlated with the initial density at k ≤0.35 h Mpc-1 . The reconstruction also reduces the power in the difference between reconstructed and initial fields by more than 2 orders of magnitude at k ≤0.2 h Mpc-1 , and it extends the range of scales where the full broadband shape of the power spectrum matches linear theory by a factor of 2-3. As a specific application, we consider measurements of the baryonic acoustic oscillation (BAO) scale that can be improved by reducing the degradation effects of large-scale flows. In our idealized dark matter simulations, the method improves the BAO signal-to-noise ratio by a factor of 2.7 at z =0 and by a factor of 2.5 at z =0.6 , improving standard BAO reconstruction by 70% at z =0 and 30% at z =0.6 , and matching the optimal BAO signal and signal-to-noise ratio of the linear density in the same volume. For BAO, the iterative nature of the reconstruction is the most important aspect.

The article deals with a classical inverse problem: the computation of the refractive index of a medium from ultrasound time-of-flight measurements. This problem is very popular in seismics but also for tomographic problems in inhomogeneous media. For example ultrasound vector field tomography needs a priori knowledge of the sound speed. According to Fermat’s principle ultrasound signals travel along geodesic curves of a Riemannian metric which is associated with the refractive index. The inverse problem thus consists of determining the index of refraction from integrals along geodesics curves associated with the integrand leading to a nonlinear problem. In this article we describe a numerical solver for this problem scheme based on an iterative minimization method for an appropriate Tikhonov functional. The outcome of the method is a stable approximation of the sought index of refraction as well as a corresponding set of geodesic curves. We prove some analytical convergence results for this method and demonstrate its performance by means of several numerical experiments. Another novelty in this article is the explicit representation of the backprojection operator for the ray transform in Riemannian geometry and its numerical realization relying on a corresponding phase function that is determined by the metric. This gives a natural extension of the conventional backprojection from 2D computerized tomography to inhomogeneous geometries. The authors dedicate this article to Prof Todd Quinto on the occasion of his 65th birthday.

A recently emerging seismic acquisition technology called simultaneous source shooting has attracted much attention from both academia and industry. The key topic in the newly developed technique is the removal of intense blending interferences caused by the simultaneous ignition of multiple airgun sources. In this paper, I propose a novel inversion strategy with multiple convex constraints to improve the deblending performance based on the projection onto convex sets (POCS) iterative framework. In the POCS iterative framework, as long as the multiple constraints are convex, the iterations are guaranteed to converge. In addition to the sparse constraint, I seek another important constraint from the untainted data. I create a blending mask in order to fully utilize the useful information hidden behind the noisy blended data. The blending mask is constructed by numerically blending a matrix with all its entries set to be one and then setting the non-one entries of the blended matrix zero. I use both synthetic and field data examples to demonstrate the successful performance of the proposed method.

Based on an equivalent two-dimensional Fornasini-Marchsini model for a batch process in industry,a closed-loop robust iterative learning fault-tolerant guaranteed cost control scheme is proposed for batch processes with actuator failures.This paper introduces relevant concepts of the fault-tolerant guaranteed cost control and formulates the robust iterative learning reliable guaranteed cost controller (ILRGCC).A significant advantage is that the proposed ILRGCC design method can be used for on-line optimization against batch-to-batch process uncertainties to realize robust tracking of set-point trajectory in time and batch-to-batch sequences.For the convenience of implementation,only measured output errors of current and previous cycles are used to design a synthetic controller for iterative learning control,consisting of dynamic output feedback plus feed-forward control.The proposed controller can not only guarantee the closed-loop convergency along time and cycle sequences but also satisfy the H∞ performance level and a cost function with upper bounds for all admissible uncertainties and any actuator failures.Sufficient conditions for the controller solution are derived in terms of linear matrix inequalities (LMIs),and design procedures,which formulate a convex optimization problem with LMI constraints,are presented.An example of injection molding is given to illustrate the effectiveness and advantages of the ILRGCC design approach.

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

In this paper, we introduce a mathematical model to describe the nanoparticles transport carried by a two-phase flow in a porous medium including gravity, capillary forces and Brownian diffusion. Nonlineariterative IMPES scheme is used to solve the flow equation, and saturation and pressure are calculated at the current iteration step and then the transport equation is soved implicitly. Therefore, once the nanoparticles concentration is computed, the two equations of volume of the nanoparticles available on the pore surfaces and the volume of the nanoparticles entrapped in pore throats are solved implicitly. The porosity and the permeability variations are updated at each time step after each iteration loop. Two numerical examples, namely, regular heterogeneous permeability and random permeability are considered. We monitor the changing of the fluid and solid properties due to adding the nanoparticles. Variation of water saturation, water pressure, nanoparticles concentration and porosity are presented graph...

In this paper, we propose two iterative decision-directed methods for peak-to-average power ratio (PAPR) reduction in optical-orthogonal frequency division multiplexing (O-OFDM) systems. The proposed methods are applicable to state-of-the-art intensity modulation-direct detection (IM-DD) O-OFDM techniques for optical wireless communication (OWC) systems, including both direct-current (DC) biased O-OFDM (DCO-OFDM), and asymmetrically clipped O-OFDM (ACO-OFDM). Conventional O-OFDM suffers from high power consumption due to high PAPR. The high PAPR of the O-OFDM signal can be counteracted by clipping the signal to a predefined threshold. However, because of clipping an inevitable distortion occurs due to the loss of useful information, thus, clipping mitigation methods are required. The proposed iterative decision-directed methods operate at the receiver, and recover the lost information by mitigating the clipping distortion. Simulation results acknowledge that the high PAPR of O-OFDM can be significantly reduced using clipping, and the proposed methods can successfully circumvent the clipping distortions. Furthermore, the proposed PAPR reduction methods exhibit a much lower computational complexity compared to standard PAPR reduction methods.

A simple scheme for crack localization is discussed that is relevant to nonlinear modal tomography based on the cross-modulation of two signals at different frequencies. The scheme is illustrated by a theoretical model, in which a thin plate or bar with a single crack is excited by a strong low-frequency wave and a high-frequency probing wave (ultrasound). The crack is assumed to be small relative to all wavelengths. Nonlinear scattering from the crack is studied using a general matrix approach as well as simplified models allowing one to find the nonlinear part of crack volume variations under the given stress and then the combinational wave components in the tested material. The nonlinear response strongly depends on the crack position with respect to the peaks or nodes of the corresponding interacting signals which can be used for determination of the crack position. Juxtaposing various resonant modes interacting at the crack it is possible to retrieve both crack location and orientation. Some aspects of inverse problem solutions are also discussed, and preliminary experimental results are presented.

Full Text Available In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM. The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.

Based on the analysis of the single-stage classification results obtained by the multitemporal SPOT 5 and Landsat 7 ETM + multispectral images separately and the derived variogram texture, the best data combinations for each land cover class are selected, and the hierarchical iterative classification is then applied for land cover mapping. The proposed classification method combines the multitemporal images of different resolutions with the image texture, which can greatly improve the classification accuracy. The method and strategies proposed in the study can be easily transferred to other similar applications.

Full Text Available We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2, and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

A new method is proposed to derive the size distribution of aerosol from the simulated multiwavelength lidar extinction coefficients. The basis for this iteration is to consider the extinction efficiency factor of particles as a set of weighting function covering the entire radius region of a distribution. The weighting functions are calculated exactly from Mie theory. This method extends the inversion region by subtracting some extinction coefficient. The radius range of simulated size distribution is 0.1-10.0μm, the inversion radius range is 0.1-2.0μm, but the inverted size distributions are in good agreement with the simulated one.

To study the non-linear fracture, a non-linear constitutive model for piezoelectric ceramics was proposed, in which the polarization switching and saturation were taken into account. Based on the model, the non-linear fracture analysis was implemented using reproducing kernel particle method (RKPM). Using local J-integral as a fracture criterion, a relation curve of fracture loads against electric fields was obtained. Qualitatively, the curve is in agreement with the experimental observations reported in literature. The reproducing equation, the shape function of RKPM, and the transformation method to impose essential boundary conditions for meshless methods were also introduced. The computation was implemented using object-oriented programming method.

The fault diagnosis based on nonlinear spectral analysis is a new technique for the nonlinear fault diagnosis, but its online application could be limited because of the enormous compution requirements for the estimation of general frequency response functions. Based on the fully decoupled Volterra identification algorithm, a new online fault diagnosis method based on nonlinear spectral analysis is presented, which can availably reduce the online compution requirements of general frequency response functions. The composition and working principle of the method are described, the test experiments have been done for damping spring of a vehicle suspension system by utilizing the new method, and the results indicate that the method is efficient.

This paper presents an iterative coil sensitivity estimation method for multi-coil MRI systems. The proposed method works with coil images in the magnitude image domain. It determines a region of support (RoS), a region being composed of the same type of tissues, by a region growing algorithm, which makes use of both intensities and intensity gradients of pixels. By repeating this procedure, it can determine multiple regions of support, which together cover most of the concerned image area. The union of these regions of support provides a rough estimate of the sensitivity of each coil through dividing the intensities of pixels by the average intensity inside every region of support. The obtained rough coil sensitivity estimate is further approached with the product of multiple low-order polynomials, rather than a single one. The product of these polynomials provides a smooth estimate of the sensitivity of each coil. With the obtained sensitivities of coils, it can produce a better reconstructed image, which determines more correct regions of support and yields preciser estimates of the sensitivities of coils. In other words, the method can be iteratively implemented to improve the estimation performance. The proposed method was verified through both simulated data and clinical data from different body parts. The experimental results confirm the superiority of our method to some conventional methods.

An iterationmethod has been developed for determining crack growth and fracture resistance curve (J-R curve) from the load versus load-line displacement record only. In this method, the hardening curve, the load versus displacement curve at a given crack length, is assumed to be a power-law function, where the exponent varies with the crack length. The exponent is determined by an interative calculation method with the assumption that the exponent varies linearly with the load-line displacement. The proposed method was applied to the static J-R tests using compact tension (CT) specimens, a three-point bend (TPB) specimen, and a cracked round bar (CRB) specimen as well as it was applied to the quasi-dynamic J-R tests using CT specimens. The J-R curves determined by the proposed method were compared with those obtained by the conventional testing methodologies. The results showed that the J-R curves could be determined directly by the proposed iterationmethod with sufficient accuracy in the specimens from SA508, SA533, and SA516 pressure vessel steels and SA312 Type 347 stainless steel.

A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicularly to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiative intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiative intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

Measuring toxicity is one of the main steps in drug development. Hence, there is a high demand for computational models to predict the toxicity effects of the potential drugs. In this study, we used a dataset, which consists of four toxicity effects:mutagenic, tumorigenic, irritant and reproductive effects. The proposed model consists of three phases. In the first phase, rough set-based methods are used to select the most discriminative features for reducing the classification time and improving the classification performance. Due to the imbalanced class distribution, in the second phase, different sampling methods such as Random Under-Sampling, Random Over-Sampling and Synthetic Minority Oversampling Technique are used to solve the problem of imbalanced datasets. ITerative Sampling (ITS) method is proposed to avoid the limitations of those methods. ITS method has two steps. The first step (sampling step) iteratively modifies the prior distribution of the minority and majority classes. In the second step, a data cleaning method is used to remove the overlapping that is produced from the first step. In the third phase, Bagging classifier is used to classify an unknown drug into toxic or non-toxic. The experimental results proved that the proposed model performed well in classifying the unknown samples according to all toxic effects in the imbalanced datasets.

A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlineariterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical example confirms that the two-grid method is more efficient than that of characteristics finite-element method.

An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation, and successfully constructed some explicit and exact travelling wave solutions.

The objective of this research is to develop efficient spatial homogenization methods for coarse-mesh nodal analysis of the light water reactors in which the reference solutions are not known. The methods developed are the global/local iterative procedures, including procedures based on variational principles. The nodal expansion method (NEM) with generalized equivalence theory is employed in coarse-mesh nodal analysis. The finite difference method (FDM) is used in fine-mesh local assembly calculation. To achieve fast and stable convergence in local assembly calculation, the mixed boundary condition is imposed at the assembly surface, where the surface flux is modulated. The assembly wise fundamental mode eigenfunction is used as the modulating function. Two direct methods are developed for the global/local iterative homogenization : G{sub 1} and G{sub 2}·G{sub 1} procedure is based on the rigorous definition of the flux-weighted constants (FWCs) and G{sub 2} procedure preserves the reaction rate ratio. Three variational principles are also proposed for the assembly homogenization. The basic form is inferred from the Pomraning's variational principle. Since the two variational methods, F{sub 0} and F{sub 2}, are based on the ratio of reaction rates, these are insensitive to the amplitude of the flux and hence they are of the Lagrangian form. On the while, the other variational principle F{sub 1} is based on the reaction rate and this requires a normalization due to its property that is sensitive to the amplitude of the flux. Thus the resulting form of F{sub 1} becomes the Swinger type. The homogenization methods developed were applied to the LWR problems. In the PWR problems we treated, there is no strong need for a global/local iterative homogenization procedure, since the heterogeneity between the fuel assemblies is relatively weak. Using the assembly discontinuity factor(ADF), the nodal analysis was improved with reasonable accuracy, while no significant

This paper is concerned with a mathematical derivation of the nonlinear time-discrete Technology-Emissions Means (TEM-) model. A detailed introduction to the dynamics modelling a Joint Implementation Program concerning Kyoto Protocol is given at the end of the paper. As the nonlinear time-discrete dynamics tends to chaotic behaviour, the necessary introduction of control parameters in the dynamics of the TEM model leads to new results in the field of time-discrete control systems. Furthermore the numerical results give new insights into a Joint-Implementation Program and herewith, they may improve this important economic tool. The iterative solution presented at the end might be a useful orientation to anticipate and support Kyoto Process.

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially effcient.

In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially efficient.

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least-squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from x-ray CT projection data.

Full Text Available We consider the data fitting problem, that is, the problem of approximating a function of several variables, given by tabulated data, and the corresponding problem for inconsistent (overdetermined systems of linear algebraic equations. Such problems, connected with measurement of physical quantities, arise, for example, in physics, engineering, and so forth. A traditional approach for solving these two problems is the discrete least squares data fitting method, which is based on discrete l2-norm. In this paper, an alternative approach is proposed: with each of these problems, we associate a nondifferentiable (nonsmooth unconstrained minimization problem with an objective function, based on discrete l1- and/or l∞-norm, respectively; that is, these two norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residual using these two norms. Respective subgradients are calculated, and a subgradient method is used for solving these two problems. The emphasis is on implementation of the proposed approach. Some computational results, obtained by an appropriate iterativemethod, are given at the end of the paper. These results are compared with the results, obtained by the iterative gradient method for the corresponding “differentiable” discrete least squares problems, that is, approximation problems based on discrete l2-norm.

Full Text Available Application process of variational iterationmethod is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay function θ(t vanishes inside the integral limits such that θ(t=qt for 0

A simple one-dimensional model of CH$_3$OH maser is considered. Two techniques are used for the calculation of molecule level populations: the accelerated lambda iteration (ALI) method and the large velocity gradient (LVG), or Sobolev, approximation. The LVG approximation gives accurate results provided that the characteristic dimensions of the medium are larger than 5-10 lengths of the resonance region. We presume that this condition can be satisfied only for the largest observed maser spot distributions. Factors controlling the pumping of class I and class II methanol masers are considered.

We propose a new approximation scheme to obtain analytic expressions for the Schroedinger equation with isotropic velocity-dependent potential to determine the scattering phase shift. In order to test the validity of our approach, we applied it to an exactly solvable model for nucleon-nucleon scattering. The results of the variation iterationmethod (VIM) formalism compare quite well with those of the exactly solvable model. The developed formalism can be applied in problems concerning pion-nucleon, nucleon-nucleon, and electron-atom scattering. (orig.)

A new method to the problem of fairing planar cubic B-spline curves is introduced in this paper. The method is based on weighted progressive iterative approximation (WPIA for short) and consists of following steps:finding the bad point which needs to fair, deleting the bad point, re-inserting a new data point to keep the structure of the curve and applying WPIA method with the new set of the data points to obtain the faired curve. The new set of the data points is formed by the rest of the original data points and the new inserted point. The method can be used for shape design and data processing. Numerical examples are provided to demonstrate the effectiveness of the method.

Full Text Available One of the serious problems in any wireless communication system using multi carrier modulation technique like Orthogonal Frequency Division Multiplexing (OFDM is its Peak to Average Power Ratio (PAPR.It limits the transmission power due to the limitation of dynamic range of Analog to Digital Converter and Digital to Analog Converter (ADC/DAC and power amplifiers at the transmitter, which in turn sets the limit over maximum achievable rate.This issue is especially important for mobile terminals to sustain longer battery life time. Therefore reducing PAPR can be regarded as an important issue to realize efficient and affordable mobile communication services.This paper presents an efficient PAPR reduction method for OFDM signal. This method is based on clipping and iterative processing. Iterative processing is performed to limit PAPR in time domain but the subtraction process of the peak that over PAPR threshold with the original signal is done in frequency domain, not in time like usual clipping technique. The results of this method is capable of reducing the PAPR significantly with minimum bit error rate (BER degradation.

Full Text Available Generalized method of moments (GMM has been widely applied for estimation of nonlinear models in economics and finance. Although generalized method of moments has good asymptotic properties under fairly moderate regularity conditions, its finite sample performance is not very well. In order to improve the finite sample performance of generalized method of moments estimators, this paper studies higher-order mean squared error of two-step efficient generalized method of moments estimators for nonlinear models. Specially, we consider a general nonlinear regression model with endogeneity and derive the higher-order asymptotic mean square error for two-step efficient generalized method of moments estimator for this model using iterative techniques and higher-order asymptotic theories. Our theoretical results allow the number of moments to grow with sample size, and are suitable for general moment restriction models, which contains conditional moment restriction models as special cases. The higher-order mean square error can be used to compare different estimators and to construct the selection criteria for improving estimator’s finite sample performance.

By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.

Many different objectives and genetic algorithms have been proposed for storage ring nonlinear dynamics performance optimization. These optimization objectives include nonlinear chromaticities and driving/detuning terms, on-momentum and off-momentum dynamic acceptance, chromatic detuning, local momentum acceptance, variation of transverse invariant, Touschek lifetime, etc. In this paper, the effectiveness of several different optimization methods and objectives are compared for the nonlinear beam dynamics optimization of the Advanced Photon Source upgrade (APS-U) lattice. The optimized solutions from these different methods are preliminarily compared in terms of the dynamic acceptance, local momentum acceptance, chromatic detuning, and other performance measures.

A numerical embedding method was proposed for solving the nonlinear optimization problem. By using the nonsmooth theory, the existence and the continuation of the following path for the corresponding homotopy equations were proved. Therefore the basic theory for the algorithm of the numerical embedding method for solving the non-linear optimization problem was established. Based on the theoretical results, a numerical embedding algorithm was designed for solving the nonlinear optimization problem, and prove its convergence carefully. Numerical experiments show that the algorithm is effective.

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present an iterativemethod, first proposed by us in Ref. [1], which allows for an efficient computation of the operator, even on large lattices. The starting point is a Krylov subspace approximation, based on the Arnoldi algorithm, for the evaluation of a generic matrix function. The efficiency of this method is spoiled when the matrix has eigenvalues close to a function discontinuity. To cure this, a small number of critical eigenvectors are added to the Krylov subspace, and two different deflation schemes are proposed in this augmented subspace. The ensuing method is then applied to the sign function of the overlap Dirac operator, for two different lattice sizes. The sign function has a discontinuity along the imaginary axis, and the numerical results show how deflation dramatically improves the efficiency of the method.

One of the advantages of the variational iterationmethod is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

Full Text Available Smart charging of electrical vehicles (EVs is critical to provide the secure and cost-effective operation for distribution systems. Three model objective functions which are minimization of total supplied power, energy costs and maximization of profits are formulated. The conventional household load is modeled as a ZIP load that consists of constant power, constant current and constant impedance components. The imbalance of distribution system, constraints on nodal voltages and thermal loadings of lines and transformers are all taken into account. Utilizing the radial operation structure of distribution system, an extended iterativemethod is proposed to greatly reduce the dimensions of optimization variables and thus improve calculation speed. Impacts of the conventional household load model on the simulation results are also investigated. Case studies on three distribution systems with 2, 14, and 141 buses are performed and analyzed. It is found that the linear constrained convex quadratic programming model is applicable at each iteration, when the conventional household load is composed of constant power and constant impedance load. However, it is not applicable when the conventional household load consists of constant current load. The accuracy and computational efficiency of the proposed method are also validated.

Full Text Available Let C be a nonempty closed convex subset of a real Hilbert space H. Let {T_{n}}:C›H be a sequence of nearly nonexpansive mappings such that F:=?_{i=1}^{?}F(T_{i}?Ø. Let V:C›H be a ?-Lipschitzian mapping and F:C›H be a L-Lipschitzian and ?-strongly monotone operator. This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence {x_{n}} converges strongly to x^{*}?F which is also the unique solution of the following variational inequality: ?0, ?x?F. As a special case, this projection method can be used to find the minimum norm solution of above variational inequality; namely, the unique solution x^{*} to the quadratic minimization problem: x^{*}=argmin_{x?F}?x?². The results here improve and extend some recent corresponding results of other authors.